3.4 Components of a Vector and Unit Vectors 65 ▸ 3.2 c o n t i n u e d Use the law of sines (Appendix B.4) to find the direction S of R measured from the northerly direction: sin b sin u B R B 35.0 km sin b sin u sin 1208 0.629 R 48.2 km b 38.9° The resultant displacement of the car is 48.2 km in a direction 38.9° west of north Finalize Does the angle b that we calculated agree with an estimate made by looking at Figure 3.11a or with an actual angle measured from the diagram using the graphical S method? Is it reasonable that the magnitude of R is larger S S S than that of both A and B ? Are the units of R correct? Although the head to tail method of adding vectors works well, it suffers from two disadvantages First, some people find using the laws of cosines and sines to be awkward Second, a triangle only results if you are adding two vectors If you are adding three or more vectors, the resulting geometric shape is usually not a triangle In Section 3.4, we explore a new method of adding vectors that will address both of these disadvantages W h at I f ? Suppose the trip were taken with the two vectors in reverse order: 35.0 km at 60.0° west of north first and then 20.0 km due north How would the magnitude and the direction of the resultant vector change? Answer They would not change The commutative law for vector addition tells us that the order of vectors in an addition is irrelevant Graphically, Figure 3.11b shows that the vectors added in the reverse order give us the same resultant vector 3.4 Components of a Vector and Unit Vectors The graphical method of adding vectors is not recommended whenever high accuracy is required or in three-dimensional problems In this section, we describe a method of adding vectors that makes use of the projections of vectors along coordinate axes These projections are called the components of the vector or its rectangular components Any vector can be completely described by its components S Consider a vector A lying in the xy plane and making an arbitrary angle u with the positive x axis as shown in S Figure 3.12a This vector can be expressed as the S sum of two other component vectors A x , which is parallel to the x axis, and A y , which is parallel to the y axis S From SFigure 3.12b, we see that the three vectors form a S right triangle and that A A A x y We shall often refer to the “components S of a vector A ,” written Ax and Ay (without the boldface notation) The compoS nent Ax represents the projection of A along the x axis, and the component Ay S represents the projection of A along the y axis These components can be positive S or negative The component Ax is positiveSif the component vector A x points in the positive x direction and is negative if A x points in the negative x direction A similar statement is made for the component Ay y y Figure 3.12 (a) A vector S S S A Ay u O a S Ax A x u O b S Ax S Ay x S A lying in the xy plane can be represented by its component vectors S S A x and A y (b) The y component S vector A y can be moved to the S right so that it adds to A x The vector sum of the component S vectors is A These three vectors form a right triangle 66 Chapter 3 Vectors Pitfall Prevention 3.2 x and y Components Equations 3.8 and 3.9 associate the cosine of the angle with the x component and the sine of the angle with the y component This association is true only because we measured the angle u with respect to the x axis, so not memorize these equations If u is measured with respect to the y axis (as in some problems), these equations will be incorrect Think about which side of the triangle containing the components is adjacent to the angle and which side is opposite and then assign the cosine and sine accordingly y Ax points left and is Ax points right and is Ay points up and is Ay points up and is Ax points left and is x Ax points right and is Ay points down and is Ay points down and is Figure 3.13 The signs Sof the c omponents of a vector A depend on the quadrant in which the vector is located From Figure 3.12 and the definition of sine andScosine, we see that cos u Ax /A and that sin u Ay /A Hence, the components of A are Ax A cos u (3.8) A y A sin u (3.9) The magnitudes of these components are the lengths of the two sides of a right triS angle with a hypotenuse of length A Therefore, the magnitude and direction of A are related to its components through the expressions A "Ax2 Ay2 u tan21 a Ay Ax (3.10) (3.11) b Notice that the signs of the components Ax and A y depend on the angle u For example, if u 120°, Ax is negative and Ay is positive If u 225°, both Ax and A y are S negative Figure 3.13 summarizes the signs of the components when A lies in the various quadrants S When solving problems, you can specify a vector A either with its components Ax and A y or with its magnitude and direction A and u Suppose you are working a physics problem that requires resolving a vector into its components In many applications, it is convenient to express the components in a coordinate system having axes that are not horizontal and vertical but that are still perpendicular to each other For example, we will consider the motion of objects sliding down inclined planes For these examples, it is often convenient to orient the x axis parallel to the plane and the y axis perpendicular to the plane Q uick Quiz 3.4 Choose the correct response to make the sentence true: A component of a vector is (a) always, (b) never, or (c) sometimes larger than the magnitude of the vector Unit Vectors Vector quantities often are expressed in terms ofy unit vectors A unit vector is a dimensionless vector having a magnitude of exactly Unit vectors are used to specify a given direction and have no other physical significance They are used solely as a bookkeeping convenience in describing a direction in space We shall use the symbols ^i, ^j, and k^ to represent unit vectors pointing in the positive x, y, and z x ˆj directions, respectively (The “hats,” or circumflexes, on the symbols are a standard ˆi notation for unit vectors.) The unit vectors ^i, ^j, and k^ form a set of mutually perpendicular vectors in a right-handed coordinate system as shown in Figure 3.14a The magnitude of each unit vector equals 1; that is, i^ 5kˆ j^ k^ S product Consider a vector A lying in the xy plane as shown in Figure 3.14b The S of the component Ax and the unit vector ^i is the component vector A x A x i^ , z y a y x ˆj ˆi S kˆ Figure 3.14 (a) The unit vectors ^i, ^j, and k^ are directed along the x, y, and z axes, respectively (b) VecS tor A Ax i^ Ay ^j lying in the xy plane has components Ax and Ay A y ˆj A x ˆi z b a y S A x 3.4 Components of a Vector and Unit Vectors 67 S S y which lies on the x axis and has magnitude Ax Likewise, A y A y j is the com Ay lying on the y axis Therefore, the unit-vector ponent vector of magnitude S notation for the vector A is S A Ax i^ A y ^j (x, y) S r (3.12) For example, consider a point lying in the xy plane and having Cartesian coordinates (x, y) as in Figure 3.15 The point can be specified by the position vector S r, which in unit-vector form is given by S r x i^ y ^j (3.13) S This notation tells us that the components of r are the coordinates x and y Now let us see how to use components to add vectors when the graphical method S S is not sufficiently accurate Suppose we wish to add vector to vector in EquaB A S tion 3.12, where vector B has components Bx and By Because of the bookkeeping convenience of the unit vectors, all we is add the x and y components separately S S S The resultant vector R A B is y ˆj x ˆi x O Figure 3.15 The point whose Cartesian coordinates are (x, y) can be represented by the position vector S r x ^i y ^j S R Ax i^ Ay j^ 1 Bx i^ By j^ or S R Ax Bx i^ 1 Ay By ^j y (3.14) S Because R R x i^ R y ^j, we see that the components of the resultant vector are R x A x Bx (3.15) R y A y By Therefore, we see that in the component method of adding vectors, we add all the x components together to find the x component of the resultant vector and use the same process for the y components We can check this addition by components with a geometric construction as shown in Figure 3.16 S The magnitude of R and the angle it makes with the x axis are obtained from its components using the relationships R "R x2 R y2 " Ax Bx 2 1 Ay By 2 tan u Ry Rx Ay By Ax B x (3.16) (3.17) At times, we need to consider situations involving motion in three component directions The extension of our methods to three-dimensional vectors is straightS S forward If A and B both have x, y, and z components, they can be expressed in the form S (3.18) A Ax i^ Ay j^ Az k^ S B Bx i^ By j^ Bz k^ (3.19) R Ax Bx i^ 1 Ay By j^ 1 Az Bz k^ (3.20) S S The sum of A and B is S Notice that Equation 3.20 differs from Equation 3.14: in Equation 3.20, the resulS tant vector also has a z component R z Az Bz If a vector R has x, y, and z components, the magnitude of the vector is R !R x2 R y2 R z2 The angle ux S that R makes with the x axis is found from the expression cos ux R x /R, with similar expressions for the angles with respect to the y and z axes The extension of our method to adding more than two vectors is also straightS S S forward For example, A B C Ax Bx Cx i^ 1 Ay By Cy j^ 1 Az Bz Cz k^ We have described adding displacement vectors in this section because these types of vectors are easy to visualize We can also add other types of Ry S By R Ay S B S A x Bx Ax Rx Figure 3.16 This geometric construction for the sum of two vectors shows the relationship between the components of the S resultant R and the components of the individual vectors Pitfall Prevention 3.3 Tangents on Calculators Equation 3.17 involves the calculation of an angle by means of a tangent function Generally, the inverse tangent function on calculators provides an angle between 290° and 190° As a consequence, if the vector you are studying lies in the second or third quadrant, the angle measured from the positive x axis will be the angle your calculator returns plus 180° 68 Chapter 3 Vectors vectors, such as velocity, force, and electric field vectors, which we will in later chapters Q uick Quiz 3.5 For which of the following vectors isSthe magnitude of the vector equal to one of theScomponents of the vector? (a) A i^ ^j S (b) B 23 ^j (c) C 15 k^ Example 3.3 The Sum of Two Vectors S S Find the sum of two displacement vectors A and B lying in the xy plane and given by S A 2.0 i^ 2.0 j^ m and Solution S B 2.0 i^ 4.0 j^ m Conceptualize You can conceptualize the situation by drawing the vectors on graph paper Draw an approximation of the expected resultant vector S Categorize We categorize this example as a simple substitution problem Comparing this expression for A with S the general expression A Ax i^ Ay j^ Az k^ , we see that Ax 2.0 m, A y 2.0 m, and Az Likewise, Bx 2.0 m, By 24.0 m, and Bz We can use a two-dimensional approach because there are no z components S S S Evaluate the components of R : S S S Rx 4.0 m Use Equation 3.16 to find the magnitude of R : Find the direction of R from Equation 3.17: S R A B 2.0 2.0 i^ m 1 2.0 4.0 j^ m Use Equation 3.14 to obtain the resultant vector R : Ry 22.0 m R "Rx2 Ry2 " 4.0 m 2 1 22.0 m 2 "20 m 4.5 m tan u Ry Rx 22.0 m 20.50 4.0 m Your calculator likely gives the answer 227° for u tan21(20.50) This answer is correct if we interpret it to mean 27° clockwise from the x axis Our standard form has been to quote the angles measured counterclockwise from the 1x axis, and that angle for this vector is u 333° Example 3.4 The Resultant Displacement A particle undergoes three consecutive displacements: DS r 15 i^ 30 j^ 12 k^ cm, DS r 23 i^ 14 j^ 5.0 k^ cm, S and D r 213 i^ 15 j^ cm Find unit-vector notation for the resultant displacement and its magnitude Solution Conceptualize Although x is sufficient to locate a point S in one dimension, we need a vector r to locate a point in S two or three dimensions The notation D r is a generalization of the one-dimensional displacement Dx in Equation 2.1 Three-dimensional displacements are more difficult to conceptualize than those in two dimensions because they cannot be drawn on paper like the latter For this problem, let us imagine that you start with your pencil at the origin of a piece of graph paper on which you have drawn x and y axes Move your pencil 15 cm to the right along the x axis, then 30 cm upward along the y axis, and then 12 cm perpendicularly toward you away from the graph paper This procedure provides the displacement described by DS r From this point, move your pencil 23 cm to the right parallel to the x axis, then 14 cm parallel to the graph paper in the 2y direction, and then 5.0 cm perpendicularly away from you toward the graph paper You are now at the displacement from the origin described by DS r 1 DS r From this point, move your pencil 13 cm to the left in the 2x direction, and (finally!) 15 cm parallel to the graph paper along the y axis Your final position is at a displacement DS r 1 DS r DS r3 from the origin 3.4 Components of a Vector and Unit Vectors 69 ▸ 3.4 c o n t i n u e d Categorize Despite the difficulty in conceptualizing in three dimensions, we can categorize this problem as a substitution problem because of the careful bookkeeping methods that we have developed for vectors The mathematical manipulation keeps track of this motion along the three perpendicular axes in an organized, compact way, as we see below DS r DS r 1 DS r DS r3 15 23 13 i^ cm 1 30 14 15 j^ cm 1 12 5.0 k^ cm 25 i^ 31 j^ 7.0 k^ cm To find the resultant displacement, add the three vectors: R "R x2 R y2 R z2 Find the magnitude of the resultant vector: " 25 cm 2 1 31 cm 2 1 7.0 cm 2 40 cm Example 3.5 Taking a Hike N A hiker begins a trip by first walking 25.0 km southeast from her car She stops and sets up her tent for the night On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower y (km) (A) D etermine the components of the hiker’s displacement for each day 20 S Conceptualize We conceptualize the problem by drawing a sketch as in Figure S 3.17 If we denote the displacement vectors on the first and second days by A and S B , respectively, and use the car as the origin of coordinates, we obtain the vectors shown in Figure 3.17 The sketch allows us to estimate the resultant vector as shown S Categorize Having drawn the resultant R , we can now categorize this problem Car 210 220 Tower S R 10 Solution E W 45.0 20 S A S B 30 40 x (km) 60.0 Tent Figure 3.17 (Example 3.5) The total displacement of the hiker is as one we’ve solved before: an addition of two vectors You should now have a S S S the vector R A B hint of the power of categorization in that many new problems are very similar to problems we have already solved if we are careful to conceptualize them Once we have drawn the displacement vectors and categorized the problem, this problem is no longer about a hiker, a walk, a car, a tent, or a tower It is a problem about vector addition, one that we have already solved S Analyze Displacement A has a magnitude of 25.0 km and is directed 45.0° below the positive x axis S Find the components of A using Equations 3.8 and 3.9: Ax A cos 245.08 25.0 km 0.707 17.7 km Ay A sin 245.08 25.0 km 20.707 217.7 km The negative value of Ay indicates that the hiker walks in the negative y direction on the first day The signs of Ax and Ay also are evident from Figure 3.17 S Find the components of B using Equations 3.8 and 3.9: Bx B cos 60.08 40.0 km 0.500 20.0 km By B sin 60.08 40.0 km 0.866 34.6 km S S (B) Determine the components of the hiker’s resultant displacement R for the trip Find an expression for R in terms of unit vectors Solution Use Equation 3.15 to find the components of the resulS S S tant displacement R A B : Rx Ax Bx 17.7 km 20.0 km 37.7 km Ry Ay By 217.7 km 34.6 km 17.0 km continued 70 Chapter 3 Vectors ▸ 3.5 c o n t i n u e d S R 37.7 i^ 17.0 j^ km Write the total displacement in unit-vector form: Finalize Looking at the graphical representation in Figure 3.17, we estimate the position of the tower to be about S (38 km, 17 km), which is consistent with the components of R in our result for the final position of the hiker Also, S both components of R are positive, putting the final position in the first quadrant of the coordinate system, which is also consistent with Figure 3.17 After reaching the tower, the hiker wishes to return to her car along a single straight line What are the components of the vector representing this hike? What should the direction of the hike be? W h at I f ? S S Answer The desired vector R car is the negative of vector R : S S R car R 237.7 i^ 17.0 j^ km The direction is found by calculating the angle that the vector makes with the x axis: tan u R car,y R car,x 217.0 km 0.450 237.7 km which gives an angle of u 204.2°, or 24.2° south of west Summary Definitions Scalar quantities are those that have only a numerical value and no associated direction Vector quantities have both magnitude and direction and obey the laws of vector addition The magnitude of a vector is always a positive number Concepts and Principles When two or more vectors are added together, they must all have the same units and they all must be the S same type of quantity We can add two vectors A and S B graphically In this method (Fig 3.6), the resultant S S S S vector R A B runs from the tail of A to the S tip of B S If a vector A has an x component Ax and a y component Ay, the vector can be expressed in unit-vector form S as A Ax i^ Ay ^j In this notation, ^i is a unit vector pointing in the positive x direction and ^j is a unit vector pointing in the positive y direction Because ^i and ^j are unit vectors, i^ j^ A second method of adding vectors involves com ponents of the vectors The x component Ax of the S S vector A is equal to the projection of A along the x axis of a coordinate system, where Ax A cos u S S The y component Ay of A is the projection of A along the y axis, where Ay A sin u We can find the resultant of two or more vectors by resolving all vectors into their x and y components, adding their resultant x and y components, and then using the Pythagorean theorem to find the magnitude of the resultant vector We can find the angle that the resultant vector makes with respect to the x axis by using a suitable trigonometric function Conceptual Questions Objective Questions 71 1. denotes answer available in Student Solutions Manual/Study Guide What is the magnitude of the vector 10 i^ 10 k^ m/s? (a) 0 (b) 10 m/s (c) 210 m/s (d) 10 (e) 14.1 m/s A vector lying in the xy plane has components of opposite sign The vector must lie in which quadrant? (a) the first quadrant (b) the second quadrant (c) the third quadrant (d) the fourth quadrant (e) either the second or the fourth quadrant S S of S the Figure OQ3.3 shows two vectors D and D Which S possibilities (a) through (d) is the vector D 2 D 1, or (e) is it none of them? S must be in which quadrant, (a) the first, (b) the second, (c) the third, or (d) the fourth, or (e) is more than one answer possible? Yes or no: Is each of the following quantities a vector? (a) force (b) temperature (c) the volume of water in a can (d) the ratings of a TV show (e) the height of a building (f) the velocity of a sports car (g) the age of the Universe What is the y component of the vector i^ k^ m/s? (a) 3 m/s (b) 28 m/s (c) (d) m/s (e) none of those answers What is the x component of the vector shown in Figure OQ3.9? (a) cm (b) cm (c) 24 cm (d) 26 cm (e) none of those answers D1 S D2 y (cm) a b c d Figure OQ3.3 The cutting tool on a lathe is given two displacements, one of magnitude cm and one of magnitude cm, in each one of five situations (a) through (e) diagrammed in Figure OQ3.4 Rank these situations according to the magnitude of the total displacement of the tool, putting the situation with the greatest resultant magnitude first If the total displacement is the same size in two situations, give those letters equal ranks a b c d e Figure OQ3.4 S TheSmagnitude of vector A is km, and the magnitude following are possible valof B is km Which of the S S ues for the magnitude of A B ? Choose all possible answers (a) 10 km (b) km (c) km (d) (e) 22 km S Let vector A point from the originSinto the second the quadrant of the xy plane and vector B point from S S origin into the fourth quadrant The vector B A Conceptual Questions 24 22 x (cm) 22 Figure OQ3.9 Objective Questions and 10 10 What is the y component of the vector shown in Figure OQ3.9? (a) cm (b) cm (c) 24 cm (d) 26 cm (e) none of those answers S 11 Vector A lies in the xy plane Both of its components will be negative if it points from the origin into which quadrant? (a) the first quadrant (b) the second quadrant (c) the third quadrant (d) the fourth quadrant (e) the second or fourth quadrants 12 A submarine dives from the water surface at an angle of 30° below the horizontal, following a straight path 50 m long How far is the submarine then below the water surface? (a) 50 m (b) (50 m)/sin 30° (c) (50 m) sin 30° (d) (50 m) cos 30° (e) none of those answers 13 A vector points from the origin into the second quadrant of the xy plane What can you conclude about its components? (a) Both components are positive (b) The x component is positive, and the y component is negative (c) The x component is negative, and the y component is positive (d) Both components are negative (e) More than one answer is possible 1. denotes answer available in Student Solutions Manual/Study Guide Is it possible to add a vector quantity to a scalar quantity? Explain Can the magnitude of a vector have a negative value? Explain A book is moved once around the perimeter of a tabletop with the dimensions 1.0 m by 2.0 m The book ends up at its initial position (a) What is its displacement? (b) What is the distance traveled? S If the component of vector A along the direction of vector S B is zero, what can you conclude about the two vectors? On a certain calculator, the inverse tangent function returns a value between 290° and 190° In what cases will this value correctly state the direction of a vector in the xy plane, by giving its angle measured counterclockwise from the positive x axis? In what cases will it be incorrect? 72 Chapter 3 Vectors Problems The problems found in this chapter may be assigned online in Enhanced WebAssign straightforward; intermediate; challenging AMT Analysis Model tutorial available in Enhanced WebAssign GP Guided Problem M Master It tutorial available in Enhanced WebAssign W Watch It video solution available in Enhanced WebAssign full solution available in the Student Solutions Manual/Study Guide BIO Q/C S Section 3.1 Coordinate Systems S The polar coordinates of a point are r 5.50 m and W u 240° What are the Cartesian coordinates of this point? The rectangular coordinates of a point are given by (2, y), and its polar coordinates are (r, 30°) Determine (a) the value of y and (b) the value of r Two points in the xy plane have Cartesian coordinates (2.00, 24.00) m and (23.00, 3.00) m Determine (a) the distance between these points and (b) their polar coordinates Two points in a plane have polar coordinates (2.50 m, W 30.0°) and (3.80 m, 120.0°) Determine (a) the Cartesian coordinates of these points and (b) the distance between them The polar coordinates of a certain point are (r 4.30 cm, u 214°) (a) Find its Cartesian coordinates x and y Find the polar coordinates of the points with Cartesian coordinates (b) (2x, y), (c) (22x, 22y), and (d) (3x, 23y) Let the polar coordinates of the point (x, y) be (r, u) S Determine the polar coordinates for the points (a) (2x, y), (b) (22x, 22y), and (c) (3x, 23y) Section 3.2 Vector and Scalar Quantities Section 3.3 Some Properties of Vectors A surveyor measures the distance across a straight river W by the following method (Fig P3.7) Starting directly across from a tree on the opposite bank, she walks d 100 m along the riverbank to establish a baseline Then she sights across to the tree The angle from her baseline to the tree is u 35.0° How wide is the river? u d Figure P3.7 S Vector A has a magnitude of 29 units and points Sin S the positive y direction When vector B is added to A , S the resultant vector A B points in the negative y direction with a magnitude of 14 units Find the magS nitude and direction of B Why is the following situation impossible? A skater glides along a circular path She defines a certain point on the circle as her origin Later on, she passes through a point at which the distance she has traveled along the path from the origin is smaller than the magnitude of her displacement vector from the origin S 10 A force F of magnitude 6.00 units acts on an object at the origin in a direction u 30.0° above the positive x axis (Fig P3.10) A S second force F of magnitude 5.00 units acts on the object in the direction of the positive y axis Find graphically the magnitude andSdirection of the resulS tant force F 1 F S F2 S F1 u Figure P3.10 S 11 The S displacement vectors A M and B shown in Figure P3.11 both have magnitudes of 3.00 Sm The direction of vec5 30.0° Find gra tor A is u S S S S A B , (b)S A S B, phically (a) S S (c) B A , and (d) A 2 B (Report all angles counterclockwise from the positive x axis.) S y S B S A u O x Figure P3.11 12 Three displacements are A Problems 11 and 22 S B 250 m Q/C 200 m due south, S due west, and C 150 m at 30.0° east of north (a) Construct a separate diagram for each of the following posS S S S A B C; sible ways of adding these vectors: R1 S S S S S S S S R B C A ; R C B A (b) Explain what you can conclude from comparing the diagrams 13 A roller-coaster car moves 200 ft horizontally and then rises 135 ft at an angle of 30.0° above the horizontal It next travels 135 ft at an angle of 40.0° downward What is its displacement from its starting point? Use graphical techniques 14 A plane flies from base camp to Lake A, 280 km away in the direction 20.0° north of east After dropping off supplies, it flies to Lake B, which is 190 km at 30.0° west of north from Lake A Graphically determine the distance and direction from Lake B to the base camp 73 Problems Section 3.4 Components of a Vector and Unit Vectors 15 A vector has an x component of 225.0 units and a y W component of 40.0 units Find the magnitude and direction of this vector this information to find the displacement from Dallas to Chicago S 16 Vector A has a magnitude of 35.0 units and points in the direction 325° counterclockwise from the positive x axis Calculate the x and y components of this vector 17 A minivan travels straight north in the right lane of a Q/C divided highway at 28.0 m/s A camper passes the minivan and then changes from the left lane into the right lane As it does so, the camper’s path on the road is a straight displacement at 8.50° east of north To avoid cutting off the minivan, the north–south distance between the camper’s back bumper and the minivan’s front bumper should not decrease (a) Can the camper be driven to satisfy this requirement? (b) Explain your answer 18 A person walks 25.0° north of east for 3.10 km How far would she have to walk due north and due east to arrive at the same location? 19 Obtain expressions in component form for the posiM tion vectors having the polar coordinates (a) 12.8 m, 150°; (b) 3.30 cm, 60.0°; and (c) 22.0 in., 215° 20 A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east (a) What is her resultant displacement? (b) What is the total distance she travels? 21 While exploring a cave, a spelunker starts at the entrance and moves the following distances in a horizontal plane She goes 75.0 m north, 250 m east, 125 m at an angle u 30.0° north of east, and 150 m south Find her resultant displacement from the cave entrance Figure P3.21 suggests the situation but is not drawn to scale u N Cave entrance W E S Final position Figure P3.21 S 22 Use S the component method to add the vectors A vectors have magand B shown in Figure P3.11 Both S of nitudes of 3.00 m and vector A makes an angle S S u 30.0° with the x axis Express the resultant A B in unit-vector notation S S ^i 2 23 Consider the two vectors A 3Si^ S2 ^j and BS 5 2S S S ^ (a) A B , (b) A B , (c)S A S B 0, M j Calculate S S 0 A B B and (e) the directions of and A , (d) S S A B 24 A map suggests that Atlanta is 730 miles in a direction of 5.00° north of east from Dallas The same map shows that Chicago is 560 miles in a direction of 21.0° west of north from Atlanta Figure P3.24 shows the locations of these three cities Modeling the Earth as flat, use Chicago 21.0 560 mi 730 mi Dallas Atlanta 5.00 Figure P3.24 25 Your dog is running around the grass in your back M yard He undergoes successive displacements 3.50 m south, 8.20 m northeast, and 15.0 m west What is the resultant displacement? S S B S 26 Given the vectors A 2.00 i^ 6.00 ^j and S S sum C A B W 3.00 i^ 2.00 j^ , (a) draw theS vector S S and theSvector dif ference D A B (b) Calculate S S C and D , in terms of unit vectors (c) Calculate C and S D in terms of polar coordinates, with angles measured with respect to the positive x axis 27 A novice golfer on the green takes three strokes to sink the ball The successive displacements of the ball are 4.00 m to the north, 2.00 m northeast, and 1.00 m at 30.0° west of south (Fig P3.27) Starting at the same initial point, an expert golfer could make the hole in what single displacement? N W 2.00 m 1.00 m E S 30.0 4.00 m Figure P3.27 A snow-covered ski slope makes an angle of 35.0° with the horizontal When a ski jumper plummets onto the hill, a parcel of splashed snow is thrown up to a maximum displacement of 1.50 m at 16.0° from the vertical in the uphill direction as shown in Figure P3.28 Find the components of its maximum displacement (a) parallel to the surface and (b) perpendicular to the surface 16.0 35.0 Figure P3.28 29 The helicopter view in Fig P3.29 (page 74) shows two mule The person on W people pulling on a stubborn S the right pulls with a force F 1 of magnitude 120 N 74 Chapter 3 Vectors and direction of u1 60.0° The person onSthe left pulls with a force F of magnitude 80.0 N and direction of u2 75.0° Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero The forces are measured in units of newtons (symbolized N) student has learned that a single equation cannot be solved to determine values for more than one unknown in it How would you explain to him that both a and b can be determined from the single equation used in part (a)? y S F2 u2 S F1 u1 x 38 Three displacement vectors of a croquet ball are S shown in Figure 0 20.0 units, A P3.38, where S S B 40.0 units, and C 30.0 units Find (a) the resultant in unitvector notation and (b) the magnitude and direction of the resultant displacement y S B S A 45.0 O x 45.0 S C 30 In a game of American footFigure P3.29 ball, a quarterback takes the ball from the line of scrimmage, runs backward a distance of 10.0 yards, and then runs sideways parallel to the line of scrimmage for 15.0 yards At this point, he throws a forward pass downfield 50.0 yards perpendicular to the line of scrimmage What is the magnitude of the football’s resultant displacement? 39 A man pushing a mop across a floor Figure P3.38 M causes it to undergo two displacements The first has a magnitude of 150 cm and makes an angle of 120° with the positive x axis The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35.0° to the positive x axis Find the magnitude and direction of the second displacement 31 Consider theS three displacement vectors A S W i^ j^ m, B i^ j^ m, and C 22 i^ j^ m Use the component methodSto determine (a) the S S S magnitude and direction of D 5 A 1 SB C Sand (b) the magnitude and direction of E A S S B C BIO and female (f) anatomies The displacements d 1m and S S 32 Vector A has x and y components of 28.70 cm and S W 15.0 cm, respectively; vector B has x and y components of 13.2 cm and 26.60 cm, respectively S S S S If A B C 0, what are the components of C ? 40 Figure P3.40 illustrates typical proportions of male (m) S S d 1f from the soles of the feet to the navel have magnitudes of 104 cm and 84.0 cm, respectively The disS S placements d 2m and d 2f from the navel to outstretched fingertips have magnitudes of 100 cm and 86.0 cm, respectively Find the vector sum of these displacements S S S d d 1 d for both people S d2m S 33 The vector A has x, y, and z components of 8.00, units, respectively (a) Write a vector M 12.0, and 24.00 S (b) Obtain a expression for A in unit-vector notation S unit-vector S expression for a vector B one-fourth the S length of A pointing in the same direction as AS (c) Obtain a unit-vector S expression for a vector C three times the length of SA pointing in the direction opposite the direction of A S d2f 23.0 28.0 S S d1m d1f S 34 Vector B has x, y, and z components of 4.00, 6.00, and 3.00 units, respectively Calculate (a) the magnitude of S S B and (b) the angle that B makes with each coordinate axis S 35 Vector A has a negative x component 3.00 units in 2.00 units in length M length and a positive y component S (a) Determine an expression for A in unit-vector notaS tion (b) Determine the magnitude and direction of A S S (c) What vector B when added to A gives a resultant vector with no x component and a negative y component 4.00 units in length? S 36 GivenSthe displacement vectors A i^ j^ k^ m W and B i^ j^ k^ m, find the magnitudes of the following vectors and express each inSterms of its S S S C A B (b) D 5 rectangular components (a) S S 2A B S S 37 (a) Taking A 6.00 i^ 8.00 j^ units, B 28.00 i^ S Q/C 3.00 j^ units, and C 26.0 i^ 19.0 j^ units, deterS S S mine a and b such that a A b B C (b) A Figure P3.40 41 Express in unit-vector notation the following vectors, S each of which has magnitude 17.0 cm (a) Vector E is directed 27.0° S counterclockwise from the positive x axis (b) Vector F is directed 27.0°Scounterclockwise from the positive y axis (c) Vector G is directed 27.0° clockwise from the negative y axis A radar station locates a sinking ship at range 17.3 km and bearing 136° clockwise from north From the same station, a rescue plane is at horizontal range 19.6 km, 153° clockwise from north, with elevation 2.20 km (a) Write the position vector for the ship relative to the plane, letting ^i represent east, ^j north, and k^ up (b) How far apart are the plane and ship? 43 Review As it passes over Grand Bahama Island, the AMT eye of a hurricane is moving in a direction 60.08 north GP of west with a speed of 41.0 km/h (a) What is the unit- vector expression for the velocity of the hurricane? 100 Chapter 4 Motion in Two Dimensions Objective Questions 1. denotes answer available in Student Solutions Manual/Study Guide Figure OQ4.1 shows a bird’s-eye view of a car going around a highway curve As the car moves from point to point 2, its speed doubles Which of the vectors (a) through (e) shows the direction of the car’s average acceleration between these two points? (a) (b) (c) (d) (e) Figure OQ4.1 Entering his dorm room, a student tosses his book bag to the right and upward at an angle of 45° with the horizontal (Fig OQ4.2) Air resistance does not affect the bag The bag moves through point A immediately after it leaves the student’s hand, through point B at the top of its flight, and through point C immediately before it lands on the top bunk bed (i) Rank the following horizontal and vertical velocity components from the largest to the smallest (a) v Ax (b) v Ay (c) v Bx (d) v By (e) v Cy Note that zero is larger than a negative number If two quantities are equal, show them as equal in your list If any quantity is equal to zero, show that fact in your list (ii) Similarly, rank the following acceleration components (a) a Ax (b) a Ay (c) a Bx (d) a By (e) a Cy B A 45Њ C Figure OQ4.2 A student throws a heavy red ball horizontally from a balcony of a tall building with an initial speed vi At the same time, a second student drops a lighter blue ball from the balcony Neglecting air resistance, which statement is true? (a) The blue ball reaches the ground first (b) The balls reach the ground at the same instant (c) The red ball reaches the ground first (d) Both balls hit the ground with the same speed (e) None of statements (a) through (d) is true A projectile is launched on the Earth with a certain initial velocity and moves without air resistance Another projectile is launched with the same initial velocity on the Moon, where the acceleration due to gravity is onesixth as large How does the maximum altitude of the projectile on the Moon compare with that of the projectile on the Earth? (a) It is one-sixth as large (b) It is the same (c) It is !6 times larger (d) It is times larger (e) It is 36 times larger Does a car moving around a circular track with constant speed have (a) zero acceleration, (b) an acceleration in the direction of its velocity, (c) an acceleration directed away from the center of its path, (d) an acceleration directed toward the center of its path, or (e) an acceleration with a direction that cannot be determined from the given information? An astronaut hits a golf ball on the Moon Which of the following quantities, if any, remain constant as a ball travels through the vacuum there? (a) speed (b) acceleration (c) horizontal component of velocity (d) vertical component of velocity (e) velocity A projectile is launched on the Earth with a certain initial velocity and moves without air resistance Another projectile is launched with the same initial velocity on the Moon, where the acceleration due to gravity is onesixth as large How does the range of the projectile on the Moon compare with that of the projectile on the Earth? (a) It is one-sixth as large (b) It is the same (c) It is !6 times larger (d) It is times larger (e) It is 36 times larger A girl, moving at m/s on in-line skates, is overtaking a boy moving at m/s as they both skate on a straight path The boy tosses a ball backward toward the girl, giving it speed 12 m/s relative to him What is the speed of the ball relative to the girl, who catches it? (a) (8 12) m/s (b) (8 2 12) m/s (c) (8 12) m/s (d) (8 12) m/s (e) (28 12) m/s A sailor drops a wrench from the top of a sailboat’s vertical mast while the boat is moving rapidly and steadily straight forward Where will the wrench hit the deck? (a) ahead of the base of the mast (b) at the base of the mast (c) behind the base of the mast (d) on the windward side of the base of the mast (e) None of the choices (a) through (d) is true 10 A baseball is thrown from the outfield toward the catcher When the ball reaches its highest point, which statement is true? (a) Its velocity and its acceleration are both zero (b) Its velocity is not zero, but its acceleration is zero (c) Its velocity is perpendicular to its acceleration (d) Its acceleration depends on the angle at which the ball was thrown (e) None of statements (a) through (d) is true 11 A set of keys on the end of a string is swung steadily in a horizontal circle In one trial, it moves at speed v in a circle of radius r In a second trial, it moves at a higher speed 4v in a circle of radius 4r In the second trial, how does the period of its motion compare with its period in the first trial? (a) It is the same as in the first trial (b) It is times larger (c) It is one-fourth as large (d) It is 16 times larger (e) It is one-sixteenth as large Problems 12 A rubber stopper on the end of a string is swung steadily in a horizontal circle In one trial, it moves at speed v in a circle of radius r In a second trial, it moves at a higher speed 3v in a circle of radius 3r In this second trial, is its acceleration (a) the same as in the first trial, (b) three times larger, (c) one-third as large, (d) nine times larger, or (e) one-ninth as large? 13 In which of the following situations is the moving object appropriately modeled as a projectile? Choose all correct answers (a) A shoe is tossed in an arbitrary Conceptual Questions 101 direction (b) A jet airplane crosses the sky with its engines thrusting the plane forward (c) A rocket leaves the launch pad (d) A rocket moves through the sky, at much less than the speed of sound, after its fuel has been used up (e) A diver throws a stone under water 14 A certain light truck can go around a curve having a radius of 150 m with a maximum speed of 32.0 m/s To have the same acceleration, at what maximum speed can it go around a curve having a radius of 75.0 m? (a) 64 m/s (b) 45 m/s (c) 32 m/s (d) 23 m/s (e) 16 m/s 1. denotes answer available in Student Solutions Manual/Study Guide A spacecraft drifts through space at a constant velocity Suddenly, a gas leak in the side of the spacecraft gives it a constant acceleration in a direction perpendicular to the initial velocity The orientation of the spacecraft does not change, so the acceleration remains perpendicular to the original direction of the velocity What is the shape of the path followed by the spacecraft in this situation? An ice skater is executing a figure eight, consisting of two identically shaped, tangent circular paths Throughout the first loop she increases her speed uniformly, and during the second loop she moves at a constant speed Draw a motion diagram showing her velocity and acceleration vectors at several points along the path of motion If you know the position vectors of a particle at two points along its path and also know the time interval during which it moved from one point to the other, can you determine the particle’s instantaneous velocity? Its average velocity? Explain Describe how a driver can steer a car traveling at constant speed so that (a) the acceleration is zero or (b) the magnitude of the acceleration remains constant A projectile is launched at some angle to the horizontal with some initial speed vi , and air resistance is negligible (a) Is the projectile a freely falling body? (b) What is its acceleration in the vertical direction? (c) What is its acceleration in the horizontal direction? Construct motion diagrams showing the velocity and acceleration of a projectile at several points along its path, assuming (a) the projectile is launched horizontally and (b) the projectile is launched at an angle u with the horizontal Explain whether or not the following particles have an acceleration: (a) a particle moving in a straight line with constant speed and (b) a particle moving around a curve with constant speed Problems The problems found in this chapter may be assigned online in Enhanced WebAssign straightforward; intermediate; challenging full solution available in the Student Solutions Manual/Study Guide AMT Analysis Model tutorial available in Enhanced WebAssign GP Guided Problem M Master It tutorial available in Enhanced WebAssign W Watch It video solution available in Enhanced WebAssign BIO Q/C S Section 4.1 The Position, Velocity, and Acceleration Vectors A motorist drives south at 20.0 m/s for 3.00 min, then turns west and travels at 25.0 m/s for 2.00 min, and finally travels northwest at 30.0 m/s for 1.00 For this 6.00-min trip, find (a) the total vector displacement, (b) the average speed, and (c) the average velocity Let the positive x axis point east When the Sun is directly overhead, a hawk dives toward the ground with a constant velocity of 5.00 m/s at 60.0° below the horizontal Calculate the speed of its shadow on the level ground Suppose the position vector for a particle is given as S a function of time by r t x t ^i y t ^j, with x(t) at b and y(t) ct d, where a 1.00 m/s, b 1.00 m, c 0.125 m/s2, and d 1.00 m (a) Calculate the average velocity during the time interval from t 2.00 s to t 4.00 s (b) Determine the velocity and the speed at t 2.00 s The coordinates of an object moving in the xy plane Q/C vary with time according to the equations x 25.00 sin vt and y 4.00 5.00 cos vt, where v is a constant, x and y are in meters, and t is in seconds (a) Determine the components of velocity of the object at t (b) Determine the components of acceleration of the object at t (c) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time t (d) Describe the path of the object in an xy plot 102 Chapter 4 Motion in Two Dimensions A golf ball is hit off a tee at the edge of a cliff Its x and y coordinates as functions of time are given by x 18.0t and y 4.00t 4.90t 2, where x and y are in meters and t is in seconds (a) Write a vector expression for the ball’s position as a function of time, using the unit vectors ^i and ^j By taking derivatives, obtain expressions for v as a function of time and (b) the velocity vector S a as a function of time (c) the acceleration vector S (d) Next use unit-vector notation to write expressions for the position, the velocity, and the acceleration of the golf ball at t 3.00 s Section 4.2 Two-Dimensional Motion with Constant Acceleration A particle initially located at the origin has an accelS W eration of a 3.00 ^j m/s and an initial velocity of S ^ vi 5.00i m/s Find (a) the vector position of the particle at any time t, (b) the velocity of the particle at any time t, (c) the coordinates of the particle at t 2.00 s, and (d) the speed of the particle at t 2.00 s The vector position of a particle varies in time accordS S W ing to the expression r 3.00i^ 6.00t ^j, where r is in meters and t is in seconds (a) Find an expression for the velocity of the particle as a function of time (b) Determine the acceleration of the particle as a function of time (c) Calculate the particle’s position and velocity at t 1.00 s It is not possible to see very small objects, such as BIO viruses, using an ordinary light microscope An elec- tron microscope, however, can view such objects using an electron beam instead of a light beam Electron microscopy has proved invaluable for investigations of viruses, cell membranes and subcellular structures, bacterial surfaces, visual receptors, chloroplasts, and the contractile properties of muscles The “lenses” of an electron microscope consist of electric and magnetic fields that control the electron beam As an example of the manipulation of an electron beam, consider an electron traveling away from the origin along the S x axis in the xy plane with initial velocity vi v i ^i As it passes through the region x to x d, the electron S experiences acceleration a a x ^i a y ^j, where ax and ay are constants For the case vi 1.80 107 m/s, ax 8.00 1014 m/s2, and ay 1.60 × 1015 m/s2, determine at x d 0.010 m (a) the position of the electron, (b) the velocity of the electron, (c) the speed of the electron, and (d) the direction of travel of the electron (i.e., the angle between its velocity and the x axis) A fish swimming in a horizontal plane has velocS AMT ity v i 4.00i^ 1.00 ^j m/s at a point in the M ocean where the position relative to a certain rock After the fish swims is S r i 10.0 ^i 4.00 ^j m with constant acceleration for 20.0 s, its velocity is S v 20.0 ^i 5.00 ^j m/s (a) What are the components of the acceleration of the fish? (b) What is the direction of its acceleration with respect to unit vector ^i ? (c) If the fish maintains constant acceleration, where is it at t 25.0 s and in what direction is it moving? 10 Review A snowmobile is originally at the point with position vector 29.0 m at 95.0° counterclockwise from the x axis, moving with velocity 4.50 m/s at 40.0° It moves with constant acceleration 1.90 m/s2 at 200° After 5.00 s have elapsed, find (a) its velocity and (b) its position vector Section 4.3 Projectile Motion Note: Ignore air resistance in all problems and take g 9.80 m/s2 at the Earth’s surface 11 Mayan kings and many school sports teams are named BIO for the puma, cougar, or mountain lion—Felis concolor— the best jumper among animals It can jump to a height of 12.0 ft when leaving the ground at an angle of 45.0° With what speed, in SI units, does it leave the ground to make this leap? 12 An astronaut on a strange planet finds that she can jump a maximum horizontal distance of 15.0 m if her initial speed is 3.00 m/s What is the free-fall acceleration on the planet? 13 In a local bar, a customer slides an empty beer mug down AMT the counter for a refill The height of the counter is M 1.22 m The mug slides off the counter and strikes the floor 1.40 m from the base of the counter (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor? 14 In a local bar, a customer slides an empty beer mug S down the counter for a refill The height of the counter is h The mug slides off the counter and strikes the floor at distance d from the base of the counter (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor? 15 A projectile is fired in such a way that its horizontal range is equal to three times its maximum height What is the angle of projection? 16 To start an avalanche on a mountain slope, an artillery W shell is fired with an initial velocity of 300 m/s at 55.0° above the horizontal It explodes on the mountainside 42.0 s after firing What are the x and y coordinates of the shell where it explodes, relative to its firing point? 17 Chinook salmon are able to move through water espe- BIO cially fast by jumping out of the water periodically This behavior is called porpoising Suppose a salmon swimming in still water jumps out of the water with velocity 6.26 m/s at 45.0° above the horizontal, sails through the air a distance L before returning to the water, and then swims the same distance L underwater in a straight, horizontal line with velocity 3.58 m/s before jumping out again (a) Determine the average velocity of the fish for the entire process of jumping and swimming underwater (b) Consider the time interval required to travel the entire distance of 2L By what percentage is this time interval reduced by the jumping/swimming process compared with simply swimming underwater at 3.58 m/s? 18 A rock is thrown upward from level ground in such a Q/C way that the maximum height of its flight is equal to S its horizontal range R (a) At what angle u is the rock thrown? (b) In terms of its original range R, what is the range R max the rock can attain if it is launched at 103 Problems of mass, which we will define in Chapter His center of mass is at elevation 1.02 m when he leaves the floor It reaches a maximum height of 1.85 m above the floor and is at elevation 0.900 m when he touches down again Determine (a) his time of flight (his “hang time”), (b) his horizontal and (c) vertical velocity components at the instant of takeoff, and (d) his takeoff angle (e) For comparison, determine the hang time of a whitetail deer making a jump (Fig P4.24b) with center-of-mass elevations yi 1.20 m, ymax 2.50 m, and yf 0.700 m 21 A firefighter, a distance d from a burning building, S directs a stream of water from a fire hose at angle ui above the horizontal as shown in Figure P4.21 If the initial speed of the stream is vi , at what height h does the water strike the building? h S vi ui d Figure P4.21 22 A landscape architect is planning an artificial waterfall in a city park Water flowing at 1.70 m/s will h leave the end of a horizontal channel at the top of a vertical wall h 2.35 m high, and from there it will fall into a pool (Fig P4.22) (a) Will the space behind Figure P4.22 the waterfall be wide enough for a pedestrian walkway? (b) To sell her plan to the city council, the architect wants to build a model to standard scale, which is one-twelfth actual size How fast should the water flow in the channel in the model? 23 A placekicker must kick a football from a point 36.0 m AMT (about 40 yards) from the goal Half the crowd hopes M the ball will clear the crossbar, which is 3.05 m high When kicked, the ball leaves the ground with a speed of 20.0 m/s at an angle of 53.0° to the horizontal (a) By how much does the ball clear or fall short of clearing the crossbar? (b) Does the ball approach the crossbar while still rising or while falling? 24 A basketball star covers 2.80 m horizontally in a jump to dunk the ball (Fig P4.24a) His motion through space can be modeled precisely as that of a particle at his center a B G Smith/Shutterstock.com © epa european pressphoto agency b.v./ Alamy the same speed but at the optimal angle for maximum range? (c) What If? Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain 19 The speed of a projectile when it reaches its maximum height is one-half its speed when it is at half its maximum height What is the initial projection angle of the projectile? 20 A ball is tossed from an upper-story window of a buildW ing The ball is given an initial velocity of 8.00 m/s at an angle of 20.0° below the horizontal It strikes the ground 3.00 s later (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown (c) How long does it take the ball to reach a point 10.0 m below the level of launching? b Figure P4.24 25 A playground is on the flat roof of a city school, 6.00 m above the street below (Fig P4.25) The vertical wall of the building is h 7.00 m high, forming a 1-m-high railing around the playground A ball has fallen to the street below, and a passerby returns it by launching it at an angle of u 53.0° above the horizontal at a point d 24.0 m from the base of the building wall The ball takes 2.20 s to reach a point vertically above the wall (a) Find the speed at which the ball was launched (b) Find the vertical distance by which the ball clears the wall (c) Find the horizontal distance from the wall to the point on the roof where the ball lands h u d Figure P4.25 26 The motion of a human body through space can be modeled as the motion of a particle at the body’s center of mass as we will study in Chapter The components of the displacement of an athlete’s center of mass from the beginning to the end of a certain jump are described by the equations xf (11.2 m/s)(cos 18.5°)t 0.360 m 0.840 m 1 11.2 m/s sin 18.58 t 12 9.80 m/s2 t where t is in seconds and is the time at which the athlete ends the jump Identify (a) the athlete’s position and (b) his vector velocity at the takeoff point (c) How far did he jump? 27 A soccer player kicks a rock horizontally off a W 40.0-m-high cliff into a pool of water If the player Motion in Two Dimensions hears the sound of the splash 3.00 s later, what was the initial speed given to the rock? Assume the speed of sound in air is 343 m/s A projectile is fired from the top of a cliff of height h S above the ocean below The projectile is fired at an angle u above the horizontal and with an initial speed vi (a) Find a symbolic expression in terms of the variables vi , g, and u for the time at which the projectile reaches its maximum height (b) Using the result of part (a), find an expression for the maximum height h max above the ocean attained by the projectile in terms of h, vi , g, and u y 29 A student stands at the GP edge of a cliff and throws S vi a stone horizontally over the edge with a speed of vi 18.0 m/s The cliff is S h 50.0 m above a body g h of water as shown in Figure P4.29 (a) What are the coordinates of the initial position of the stone? (b) What are the compox nents of the initial velocity of the stone? (c) What is the S v appropriate analysis model Figure P4.29 for the vertical motion of the stone? (d) What is the appropriate analysis model for the horizontal motion of the stone? (e) Write symbolic equations for the x and y components of the velocity of the stone as a function of time (f) Write symbolic equations for the position of the stone as a function of time (g) How long after being released does the stone strike the water below the cliff? (h) With what speed and angle of impact does the stone land? 30 The record distance in the sport of throwing cowpats Q/C is 81.1 m This record toss was set by Steve Urner of the United States in 1981 Assuming the initial launch angle was 45° and neglecting air resistance, determine (a) the initial speed of the projectile and (b) the total time interval the projectile was in flight (c) How would the answers change if the range were the same but the launch angle were greater than 45°? Explain 31 A boy stands on a diving board and tosses a stone into a swimming pool The stone is thrown from a height of 2.50 m above the water surface with a velocity of 4.00 m/s at an angle of 60.0° above the horizontal As the stone strikes the water surface, it immediately slows down to exactly half the speed it had when it struck the water and maintains that speed while in the water After the stone enters the water, it moves in a straight line in the direction of the velocity it had when it struck the water If the pool is 3.00 m deep, how much time elapses between when the stone is thrown and when it strikes the bottom of the pool? 32 A home run is hit in such a way that the baseball just M clears a wall 21.0 m high, located 130 m from home plate The ball is hit at an angle of 35.0° to the horizontal, and air resistance is negligible Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall (Assume the ball is hit at a height of 1.00 m above the ground.) Section 4.4 Analysis Model: Particle in Uniform Circular Motion Note: Problems and 13 in Chapter can also be assigned with this section 33 The athlete shown in Figure P4.33 rotates a 1.00-kg discus along a circular path of radius 1.06 m The maximum speed of the discus is 20.0 m/s Determine the magnitude of the maximum radial acceleration of the discus Adrian Dennis/AFP/Getty Images 104 Chapter 4 Figure P4.33 In Example 4.6, we found the centripetal acceleration of the Earth as it revolves around the Sun From information on the endpapers of this book, compute the centripetal acceleration of a point on the surface of the Earth at the equator caused by the rotation of the Earth about its axis 35 Casting molten metal is important in many industrial processes Centrifugal casting is used for manufacturing pipes, bearings, and many other structures A variety of sophisticated techniques have been invented, but the basic idea is as illustrated in Figure P4.35 A cylindrical enclosure is rotated rapidly and steadily about a horizontal axis Molten metal is poured into the rotating cylinder and then cooled, forming the finished product Turning the cylinder at a high rotation rate forces the solidifying metal strongly to the outside Any bubbles are displaced toward the axis, so unwanted voids will not be present in the casting Sometimes it is desirable to form a composite casting, such as for a bearing Here a strong steel outer surface is poured and then inside it a lining of special low-friction metal In some applications, a very strong metal is given a coating of corrosion-resistant metal Centrifugal casting results in strong bonding between the layers Preheated steel sheath Axis of rotation Molten metal Figure P4.35 Problems 105 Suppose a copper sleeve of inner radius 2.10 cm and outer radius 2.20 cm is to be cast To eliminate bubbles and give high structural integrity, the centripetal acceleration of each bit of metal should be at least 100g What rate of rotation is required? State the answer in revolutions per minute 41 A train slows down as it rounds a sharp horizontal M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s it takes to round the bend The radius of the curve is 150 m Compute the acceleration at the moment the train speed reaches 50.0 km/h Assume the train continues to slow down at this time at the same rate 36 A tire 0.500 m in radius rotates at a constant rate of 200 rev/min Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge) 42 A ball swings counterclockwise in a vertical circle at the end of a rope 1.50 m long When the ball is 36.9° past the lowest point on its way up, its total acceleration is 222.5 i^ 20.2 j^ m/s2 For that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball 37 Review The 20-g centrifuge at NASA’s Ames Research AMT Center in Mountain View, California, is a horizontal, cylindrical tube 58.0 ft long and is represented in Figure P4.37 Assume an astronaut in training sits in a seat at one end, facing the axis of rotation 29.0 ft away Determine the rotation rate, in revolutions per second, required to give the astronaut a centripetal acceleration of 20.0g 43 (a) Can a particle moving with instantaneous speed 3.00 m/s on a path with radius of curvature 2.00 m have an acceleration of magnitude 6.00 m/s2? (b) Can it have an acceleration of magnitude 4.00 m/s2? In each case, if the answer is yes, explain how it can happen; if the answer is no, explain why not 29 ft Section 4.6 Relative Velocity and Relative Acceleration 4 The pilot of an airplane notes that the compass indicates a heading due west The airplane’s speed relative to the air is 150 km/h The air is moving in a wind at 30.0 km/h toward the north Find the velocity of the airplane relative to the ground Figure P4.37 38 An athlete swings a ball, connected to the end of a chain, in a horizontal circle The athlete is able to rotate the ball at the rate of 8.00 rev/s when the length of the chain is 0.600 m When he increases the length to 0.900 m, he is able to rotate the ball only 6.00 rev/s (a) Which rate of rotation gives the greater speed for the ball? (b) What is the centripetal acceleration of the ball at 8.00 rev/s? (c) What is the centripetal acceleration at 6.00 rev/s? Section 4.5 Tangential and Radial Acceleration 40 Figure P4.40 represents the W total acceleration of a particle moving clockwise in a circle of radius 2.50 m at a certain instant of time For that instant, find (a) the radial acceleration of the particle, (b) the speed of the particle, and (c) its tangential acceleration a ϭ 15.0 m/s2 S v S a 2.50 m 30.0Њ Figure P4.40 NASA 39 The astronaut orbiting the Earth in Figure P4.39 is preparing to dock with a Westar VI satellite The satellite is in a circular orbit 600 km above the Earth’s surface, where the free-fall acceleration is 8.21 m/s2 Take Figure P4.39 the radius of the Earth as 400 km Determine the speed of the satellite and the time interval required to complete one orbit around the Earth, which is the period of the satellite 45 An airplane maintains a speed of 630 km/h relative to the air it is flying through as it makes a trip to a city 750 km away to the north (a) What time interval is required for the trip if the plane flies through a headwind blowing at 35.0 km/h toward the south? (b) What time interval is required if there is a tailwind with the same speed? (c) What time interval is required if there is a crosswind blowing at 35.0 km/h to the east relative to the ground? 46 A moving beltway at an airport has a speed v and a S length L A woman stands on the beltway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the beltway with a speed of v relative to the moving beltway (a) What time interval is required for the woman to travel the distance L? (b) What time interval is required for the man to travel this distance? (c) A second beltway is located next to the first one It is identical to the first one but moves in the opposite direction at speed v1 Just as the man steps onto the beginning of the beltway and begins to walk at speed v relative to his beltway, a child steps on the other end of the adjacent beltway The child stands at rest relative to this second beltway How long after stepping on the beltway does the man pass the child? 47 A police car traveling at 95.0 km/h is traveling west, chasing a motorist traveling at 80.0 km/h (a) What is the velocity of the motorist relative to the police car? (b) What is the velocity of the police car relative to the motorist? (c) If they are originally 250 m apart, in what time interval will the police car overtake the motorist? 48 A car travels due east with a speed of 50.0 km/h RainM drops are falling at a constant speed vertically with Motion in Two Dimensions respect to the Earth The traces of the rain on the side windows of the car make an angle of 60.0° with the vertical Find the velocity of the rain with respect to (a) the car and (b) the Earth 49 A bolt drops from the ceiling of a moving train car Q/C that is accelerating northward at a rate of 2.50 m/s2 (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer inside the train car (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth 50 A river has a steady speed of 0.500 m/s A student swims M upstream a distance of 1.00 km and swims back to the Q/C starting point (a) If the student can swim at a speed of 1.20 m/s in still water, how long does the trip take? (b) How much time is required in still water for the same length swim? (c) Intuitively, why does the swim take longer when there is a current? 51 A river flows with a steady speed v A student swims Q/C upstream a distance d and then back to the starting S point The student can swim at speed c in still water (a) In terms of d, v, and c, what time interval is required for the round trip? (b) What time interval would be required if the water were still? (c) Which time interval is larger? Explain whether it is always larger 52 A Coast Guard cutter detects an unidentified ship at a distance of 20.0 km in the direction 15.0° east of north The ship is traveling at 26.0 km/h on a course at 40.0° east of north The Coast Guard wishes to send a speedboat to intercept and investigate the vessel If the speedboat travels at 50.0 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north 53 A science student is riding on a flatcar of a train travAMT eling along a straight, horizontal track at a constant M speed of 10.0 m/s The student throws a ball into the air along a path that he judges to make an initial angle of 60.0° with the horizontal and to be in line with the track The student’s professor, who is standing on the ground nearby, observes the ball to rise vertically How high does she see the ball rise? 54 A farm truck moves Q/C due east with a constant S v velocity of 9.50 m/s on a limitless, horizontal stretch of road A boy riding on the back of the truck Figure P4.54 throws a can of soda upward (Fig P4.54) and catches the projectile at the same location on the truck bed, but 16.0 m farther down the road (a) In the frame of reference of the truck, at what angle to the vertical does the boy throw the can? (b) What is the initial speed of the can relative to the truck? (c) What is the shape of the can’s trajectory as seen by the boy? An observer on the ground watches the boy throw the can and catch it In this observer’s frame of reference, (d) describe the shape of the can’s path and (e) determine the initial velocity of the can Additional Problems 55 A ball on the end of a string is whirled around in a horizontal circle of radius 0.300 m The plane of the circle is 1.20 m above the ground The string breaks and the ball lands 2.00 m (horizontally) away from the point on the ground directly beneath the ball’s location when the string breaks Find the radial acceleration of the ball during its circular motion 56 A ball is thrown with an initial speed vi at an angle ui S with the horizontal The horizontal range of the ball is R, and the ball reaches a maximum height R/6 In terms of R and g, find (a) the time interval during which the ball is in motion, (b) the ball’s speed at the peak of its path, (c) the initial vertical component of its velocity, (d) its initial speed, and (e) the angle ui (f) Suppose the ball is thrown at the same initial speed found in (d) but at the angle appropriate for reaching the greatest height that it can Find this height (g) Suppose the ball is thrown at the same initial speed but at the angle for greatest possible range Find this maximum horizontal range 57 Why is the following situation impossible? A normally proportioned adult walks briskly along a straight line in the 1x direction, standing straight up and holding his right arm vertical and next to his body so that the arm does not swing His right hand holds a ball at his side a distance h above the floor When the ball passes above a point marked as x on the horizontal floor, he opens his fingers to release the ball from rest relative to his hand The ball strikes the ground for the first time at position x 7.00h 58 A particle starts from the origin with velocity 5i^ m/s at t and moves in the xy plane with a varying accela is in meters per eration given by S a 6!t j^ , where S second squared and t is in seconds (a) Determine the velocity of the particle as a function of time (b) Determine the position of the particle as a function of time 59 The “Vomit Comet.” In microgravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path As shown in Figure P4.59, the aircraft climbs from 24 000 ft to 31 000 ft, where Altitude (ft) 106 Chapter 4 31 000 45Њ nose high 24 000 45Њ nose low Zero g 65 Maneuver time (s) Figure P4.59 107 Problems it enters a parabolic path with a velocity of 143 m/s nose high at 45.0° and exits with velocity 143 m/s at 45.0° nose low During this portion of the flight, the aircraft and objects inside its padded cabin are in free fall; astronauts and equipment float freely as if there were no gravity What are the aircraft’s (a) speed and (b) altitude at the top of the maneuver? (c) What is the time interval spent in microgravity? leave simultaneously and drive for 2.50 h in the directions shown Car has a speed of 90.0 km/h If the cars arrive simultaneously at the lake, what is the speed of car 2? L 60 A basketball player is standing on the floor 10.0 m from the basket as in Figure P4.60 The height of the basket is 3.05 m, and he shoots the ball at a 40.0o angle with the horizontal from a height of 2.00 m (a) What is the acceleration of the basketball at the highest point in its trajectory? (b) At what speed must the player throw the basketball so that the ball goes through the hoop without striking the backboard? A 40.0Њ 3.05 m 2.00 m 10.0 m Figure P4.60 61 Lisa in her Lamborghini accelerates at the rate of 3.00i^ 2.00 j^ m/s2, while Jill in her Jaguar accelerates at 1.00i^ 3.00 j^ m/s2 They both start from rest at the origin of an xy coordinate system After 5.00 s, (a) what is Lisa’s speed with respect to Jill, (b) how far apart are they, and (c) what is Lisa’s acceleration relative to Jill? 62 A boy throws a stone horizontally from the top of a cliff S of height h toward the ocean below The stone strikes the ocean at distance d from the base of the cliff In terms of h, d, and g, find expressions for (a) the time t at which the stone lands in the ocean, (b) the initial speed of the stone, (c) the speed of the stone immediately before it reaches the ocean, and (d) the direction of the stone’s velocity immediately before it reaches the ocean 63 A flea is at point A on a horizontal turntable, 10.0 cm from the center The turntable is rotating at 33.3 rev/min in the clockwise direction The flea jumps straight up to a height of 5.00 cm At takeoff, it gives itself no horizontal velocity relative to the turntable The flea lands on the turntable at point B Choose the origin of coordinates to be at the center of the turntable and the positive x axis passing through A at the moment of takeoff Then the original position of the flea is 10.0 ^i cm (a) Find the position of point A when the flea lands (b) Find the position of point B when the flea lands Towns A and B in Figure P4.64 are 80.0 km apart A M couple arranges to drive from town A and meet a couple driving from town B at the lake, L The two couples 40.0° 80.0 km B Figure P4.64 65 A catapult launches a rocket at an angle of 53.0° above the horizontal with an initial speed of 100 m/s The rocket engine immediately starts a burn, and for 3.00 s the rocket moves along its initial line of motion with an acceleration of 30.0 m/s2 Then its engine fails, and the rocket proceeds to move in free fall Find (a) the maximum altitude reached by the rocket, (b) its total time of flight, and (c) its horizontal range 66 A cannon with a muzzle speed of 000 m/s is used to start an avalanche on a mountain slope The target is 000 m from the cannon horizontally and 800 m above the cannon At what angle, above the horizontal, should the cannon be fired? 67 Why is the following situation impossible? Albert Pujols hits a home run so that the baseball just clears the top row of bleachers, 24.0 m high, located 130 m from home plate The ball is hit at 41.7 m/s at an angle of 35.0° to the horizontal, and air resistance is negligible As some molten metal splashes, one droplet flies off to S the east with initial velocity vi at angle ui above the horizontal, and another droplet flies off to the west with the same speed at the same angle above the horizontal as shown in Figure P4.68 In terms of vi and ui , find the distance between the two droplets as a function of time S vi ui S vi ui Figure P4.68 69 An astronaut on the surface of the Moon fires a cannon to launch an experiment package, which leaves the barrel moving horizontally Assume the free-fall acceleration on the Moon is one-sixth of that on the 108 Chapter 4 Motion in Two Dimensions Earth (a) What must the muzzle speed of the package be so that it travels completely around the Moon and returns to its original location? (b) What time interval does this trip around the Moon require? 70 A pendulum with a cord of length r 1.00 m swings in a vertical plane (Fig P4.70) When the pendulum is in the two horizontal positions u u 90.0° and u 270°, its r speed is 5.00 m/s Find the S magnitude of (a) the radial g acceleration and (b) the ar tangential acceleration for f S a these positions (c) Draw vector diagrams to deterat mine the direction of the Figure P4.70 total acceleration for these two positions (d) Calculate the magnitude and direction of the total acceleration at these two positions 71 A hawk is flying horizontally at 10.0 m/s in a straight M line, 200 m above the ground A mouse it has been carrying struggles free from its talons The hawk continues on its path at the same speed for 2.00 s before attempting to retrieve its prey To accomplish the retrieval, it dives in a straight line at constant speed and recaptures the mouse 3.00 m above the ground (a) Assuming no air resistance acts on the mouse, find the diving speed of the hawk (b) What angle did the hawk make with the horizontal during its descent? (c) For what time interval did the mouse experience free fall? 72 A projectile is launched from the point (x 0, y 0), Q/C with velocity 12.0i^ 49.0 j^ m/s, at t (a) Make a r | from the oritable listing the projectile’s distance |S gin at the end of each second thereafter, for # t # 10 s Tabulating the x and y coordinates and the components of velocity vx and vy will also be useful (b) Notice that the projectile’s distance from its starting point increases with time, goes through a maximum, and starts to decrease Prove that the distance is a maximum when the position vector is perpendicular to the velocv is not perpendicular to ity Suggestion: Argue that if S S r , then |S r | must be increasing or decreasing (c) Determine the magnitude of the maximum displacement (d) Explain your method for solving part (c) 73 A spring cannon is located at the edge of a table that Q/C is 1.20 m above the floor A steel ball is launched from the cannon with speed vi at 35.0° above the horizontal (a) Find the horizontal position of the ball as a function of vi at the instant it lands on the floor We write this function as x(vi ) Evaluate x for (b) vi 0.100 m/s and for (c) vi 100 m/s (d) Assume vi is close to but not equal to zero Show that one term in the answer to part (a) dominates so that the function x(vi ) reduces to a simpler form (e) If vi is very large, what is the approximate form of x(vi )? (f) Describe the overall shape of the graph of the function x(vi ) 74 An outfielder throws a baseball to his catcher in an attempt to throw out a runner at home plate The ball bounces once before reaching the catcher Assume the angle at which the bounced ball leaves the ground is the same as the angle at which the outfielder threw it as shown in Figure P4.74, but that the ball’s speed after the bounce is one-half of what it was before the bounce (a) Assume the ball is always thrown with the same initial speed and ignore air resistance At what angle u should the fielder throw the ball to make it go the same distance D with one bounce (blue path) as a ball thrown upward at 45.0° with no bounce (green path)? (b) Determine the ratio of the time interval for the one-bounce throw to the flight time for the no-bounce throw θ 45.0° θ D Figure P4.74 75 A World War II bomber flies horizontally over level terrain with a speed of 275 m/s relative to the ground and at an altitude of 3.00 km The bombardier releases one bomb (a) How far does the bomb travel horizontally between its release and its impact on the ground? Ignore the effects of air resistance (b) The pilot maintains the plane’s original course, altitude, and speed through a storm of flak Where is the plane when the bomb hits the ground? (c) The bomb hits the target seen in the telescopic bombsight at the moment of the bomb’s release At what angle from the vertical was the bombsight set? 76 A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washed-out bridge (Fig P4.76) The quick stop causes a number of melons to fly off the truck One melon leaves the hood of the truck with an initial speed vi 10.0 m/s in the horizontal direction A cross section of the bank has the shape of the bottom half of a parabola, with its vertex at the initial location of the projected watermelon and with the equation y 16x, where x and y are meay vi ϭ 10.0 m/s x Figure P4.76 The blue dashed curve shows the parabolic shape of the bank Problems sured in meters What are the x and y coordinates of the melon when it splatters on the bank? 77 A car is parked on a steep incline, making an angle of M 37.0° below the horizontal and overlooking the ocean, when its brakes fail and it begins to roll Starting from rest at t 5 0, the car rolls down the incline with a constant acceleration of 4.00 m/s2, traveling 50.0 m to the edge of a vertical cliff The cliff is 30.0 m above the ocean Find (a) the speed of the car when it reaches the edge of the cliff, (b) the time interval elapsed when it arrives there, (c) the velocity of the car when it lands in the ocean, (d) the total time interval the car is in motion, and (e) the position of the car when it lands in the ocean, relative to the base of the cliff 78 An aging coyote cannot run BEEP fast enough to catch a roadBEEP runner He purchases on eBay a set of jet-powered roller skates, which provide a constant horizontal acceleration of 15.0 m/s2 (Fig. P4.78) The coyote starts at rest 70.0 m from the edge of a cliff at the instant the roadrunner zips past in the Figure P4.78 direction of the cliff (a) Determine the minimum constant speed the roadrunner must have to reach the cliff before the coyote At the edge of the cliff, the roadrunner escapes by making a sudden turn, while the coyote continues straight ahead The coyote’s skates remain horizontal and continue to operate while he is in flight, so his acceleration while in the air is 15.0 i^ 9.80 j^ m/s2 (b) The cliff is 100 m above the flat floor of the desert Determine how far from the base of the vertical cliff the coyote lands (c) Determine the components of the coyote’s impact velocity 79 A fisherman sets out upstream on a river His small boat, powered by an outboard motor, travels at a constant speed v in still water The water flows at a lower constant speed vw The fisherman has traveled upstream for 2.00 km when his ice chest falls out of the boat He notices that the chest is missing only after he has gone upstream for another 15.0 min At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water He catches up with the floating ice chest just as he returns to his starting point How fast is the river flowing? Solve this problem in two ways (a) First, use the Earth as a reference frame With respect to the Earth, the boat travels upstream at speed v vw and downstream at v vw (b) A second much simpler and more elegant solution is obtained by using the water as the reference frame This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets 80 Do not hurt yourself; not strike your hand against Q/C anything Within these limitations, describe what you to give your hand a large acceleration Compute an 109 order-of-magnitude estimate of this acceleration, stating the quantities you measure or estimate and their values Challenge Problems 81 A skier leaves the ramp of a ski jump with a velocity of Q/C v 10.0 m/s at u 15.0° above the horizontal as shown in Figure P4.81 The slope where she will land is inclined downward at f 50.0°, and air resistance is negligible Find (a) the distance from the end of the ramp to where the jumper lands and (b) her velocity components just before the landing (c) Explain how you think the results might be affected if air resistance were included S v u f Figure P4.81 82 Two swimmers, Chris and Sarah, start together at the S same point on the bank of a wide stream that flows with a speed v Both move at the same speed c (where c v) relative to the water Chris swims downstream a distance L and then upstream the same distance Sarah swims so that her motion relative to the Earth is perpendicular to the banks of the stream She swims the distance L and then back the same distance, with both swimmers returning to the starting point In terms of L, c, and v, find the time intervals required (a) for Chris’s round trip and (b) for Sarah’s round trip (c) Explain which swimmer returns first 83 The water in a river flows uniformly at a constant speed of 2.50 m/s between parallel banks 80.0 m apart You are to deliver a package across the river, but you can swim only at 1.50 m/s (a) If you choose to minimize the time you spend in the water, in what direction should you head? (b) How far downstream will you be carried? (c) If you choose to minimize the distance downstream that the river carries you, in what direction should you head? (d) How far downstream will you be carried? A person standing at the top of a hemispherical rock of S radius R kicks a ball (initially at rest on the top of the v i as shown in Figrock) to give it horizontal velocity S ure P4.84 (a) What must be its minimum initial speed S vi R Figure P4.84 x 110 Chapter 4 Motion in Two Dimensions if the ball is never to hit the rock after it is kicked? (b) With this initial speed, how far from the base of the rock does the ball hit the ground? 85 A dive-bomber has a velocity of 280 m/s at an angle u below the horizontal When the altitude of the aircraft is 2.15 km, it releases a bomb, which subsequently hits a target on the ground The magnitude of the displacement from the point of release of the bomb to the target is 3.25 km Find the angle u A projectile is fired up an incline (incline angle f) with an initial speed vi at an angle ui with respect to the horizontal (ui > f) as shown in Figure P4.86 (a) Show that the projectile travels a distance d up the incline, where 2v i cos ui sin u i f 2 d5 g cos f Path of the projectile S vi θi d φ Figure P4.86 S v i ϭ 250 m/s vi uH (b) For what value of ui is d a maximum, and what is that maximum value? 87 A fireworks rocket explodes at height h, the peak of S its vertical trajectory It throws out burning fragments in all directions, but all at the same speed v Pellets of solidified metal fall to the ground without air resistance Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal 8 In the What If? section of Example 4.5, it was claimed S that the maximum range of a ski jumper occurs for a launch angle u given by u 458 f where f is the angle the hill makes with the horizontal in Figure 4.14 Prove this claim by deriving the equation above 89 An enemy ship is on the east side of a mountain island as shown in Figure P4.89 The enemy ship has maneuvered to within 500 m of the 800-m-high mountain peak and can shoot projectiles with an initial speed of 250 m/s If the western shoreline is horizontally 300 m from the peak, what are the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship? 800 m uL 500 m Figure P4.89 300 m The Laws of Motion c h a p t e r 5.1 The Concept of Force 5.2 Newton’s First Law and Inertial Frames 5.3 Mass 5.4 Newton’s Second Law 5.5 The Gravitational Force and Weight 5.6 Newton’s Third Law 5.7 Analysis Models Using Newton’s Second Law 5.8 Forces of Friction In Chapters and 4, we described the motion of an object in terms of its position, velocity, and acceleration without considering what might influence that motion Now we consider that influence: Why does the motion of an object change? What might cause one object to remain at rest and another object to accelerate? Why is it generally easier to move a small object than a large object? The two main factors we need to consider are the forces acting on an object and the mass of the object In this chapter, we begin our study of dynamics by discussing the three basic laws of motion, which deal with forces and masses and were formulated more than three centuries ago by Isaac Newton A person sculls on a calm waterway The water exerts forces on the oars to accelerate the boat (Tetra Images/ Getty Images) 5.1 The Concept of Force Everyone has a basic understanding of the concept of force from everyday experience When you push your empty dinner plate away, you exert a force on it Similarly, you exert a force on a ball when you throw or kick it In these examples, the word force refers to an interaction with an object by means of muscular activity and some change in the object’s velocity Forces not always cause motion, however For example, when you are sitting, a gravitational force acts on your body and yet you remain stationary As a second example, you can push (in other words, exert a force) on a large boulder and not be able to move it What force (if any) causes the Moon to orbit the Earth? Newton answered this and related questions by stating that forces are what cause any change in the velocity of an object The Moon’s velocity changes in direction as it moves in a nearly circular 111 112 Chapter 5 The Laws of Motion Figure 5.1 Some examples of applied forces In each case, a force is exerted on the object within the boxed area Some agent in the environment external to the boxed area exerts a force on the object Contact forces a c b Field forces m Bridgeman-Giraudon/Art Resource, NY d Isaac Newton English physicist and mathematician (1642–1727) Isaac Newton was one of the most brilliant scientists in history Before the age of 30, he formulated the basic concepts and laws of mechanics, discovered the law of universal gravitation, and invented the mathematical methods of calculus As a consequence of his theories, Newton was able to explain the motions of the planets, the ebb and flow of the tides, and many special features of the motions of the Moon and the Earth He also interpreted many fundamental observations concerning the nature of light His contributions to physical theories dominated scientific thought for two centuries and remain important today M Ϫq e Iron ϩQ N S f orbit around the Earth This change in velocity is caused by the gravitational force exerted by the Earth on the Moon When a coiled spring is pulled, as in Figure 5.1a, the spring stretches When a stationary cart is pulled, as in Figure 5.1b, the cart moves When a football is kicked, as in Figure 5.1c, it is both deformed and set in motion These situations are all examples of a class of forces called contact forces That is, they involve physical contact between two objects Other examples of contact forces are the force exerted by gas molecules on the walls of a container and the force exerted by your feet on the floor Another class of forces, known as field forces, does not involve physical contact between two objects These forces act through empty space The gravitational force of attraction between two objects with mass, illustrated in Figure 5.1d, is an example of this class of force The gravitational force keeps objects bound to the Earth and the planets in orbit around the Sun Another common field force is the electric force that one electric charge exerts on another (Fig 5.1e), such as the attractive electric force between an electron and a proton that form a hydrogen atom A third example of a field force is the force a bar magnet exerts on a piece of iron (Fig 5.1f) The distinction between contact forces and field forces is not as sharp as you may have been led to believe by the previous discussion When examined at the atomic level, all the forces we classify as contact forces turn out to be caused by electric (field) forces of the type illustrated in Figure 5.1e Nevertheless, in developing models for macroscopic phenomena, it is convenient to use both classifications of forces The only known fundamental forces in nature are all field forces: (1) gravitational forces between objects, (2) electromagnetic forces between electric charges, (3) strong forces between subatomic particles, and (4) weak forces that arise in certain radioactive decay processes In classical physics, we are concerned only with gravitational and electromagnetic forces We will discuss strong and weak forces in Chapter 46 The Vector Nature of Force It is possible to use the deformation of a spring to measure force Suppose a vertical force is applied to a spring scale that has a fixed upper end as shown in Figure 5.2a The spring elongates when the force is applied, and a pointer on the scale reads the extension of the spring We can calibrate the spring by defining a S reference force F as the force that produces a pointer reading of 1.00 cm If we S now apply a different downward force F whose magnitude is twice that of the ref2 S erence force F as seen in Figure 5.2b, the pointer moves to 2.00 cm Figure 5.2c shows that the combined effect of the two collinear forces is the sum of the effects of the individual forces S S Now suppose the two forces are applied simultaneously with F downward and F horizontal as S illustrated in Figure 5.2d In this case, the pointer reads 2.24 cm The single force F that would produce this same reading is the sum of the two vecS S S tors F and F as described in Figure 5.2d That is, F !F12 F22 2.24 units, 5.2 Newton’s First Law and Inertial Frames 113 S A downward S force F2 elongates the spring 2.00 cm S S When F1 is downward and F2 is horizontal, the combination of the two forces elongates the spring by 2.24 cm A downward S force F1 elongates the spring 1.00 cm S When F1 and F2 are applied together in the same direction, the spring elongates by 3.00 cm 2 4 3 S F2 u S F1 S F S S F1 F1 S F2 Figure 5.2 The vector nature S F2 a b c d of a force is tested with a spring scale and its direction is u tan21 (20.500) 226.6° Because forces have been experimentally verified to behave as vectors, you must use the rules of vector addition to obtain the net force on an object 5.2 Newton’s First Law and Inertial Frames We begin our study of forces by imagining some physical situations involving a puck on a perfectly level air hockey table (Fig 5.3) You expect that the puck will remain stationary when it is placed gently at rest on the table Now imagine your air hockey table is located on a train moving with constant velocity along a perfectly smooth track If the puck is placed on the table, the puck again remains where it is placed If the train were to accelerate, however, the puck would start moving along the table opposite the direction of the train’s acceleration, just as a set of papers on your dashboard falls onto the floor of your car when you step on the accelerator As we saw in Section 4.6, a moving object can be observed from any number of reference frames Newton’s first law of motion, sometimes called the law of inertia, defines a special set of reference frames called inertial frames This law can be stated as follows: If an object does not interact with other objects, it is possible to identify a reference frame in which the object has zero acceleration Such a reference frame is called an inertial frame of reference When the puck is on the air hockey table located on the ground, you are observing it from an inertial reference frame; there are no horizontal interactions of the puck with any other objects, and you observe it to have zero acceleration in that direction When you are on the train moving at constant velocity, you are also observing the puck from an inertial reference frame Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame When you and the train accelerate, however, you are observing the puck from a noninertial reference frame because the train is accelerating relative to the inertial reference frame of the Earth’s surface While the puck appears to be accelerating according to your observations, a reference frame can be identified in which the puck has zero acceleration Airflow Electric blower Figure 5.3 On an air hockey table, air blown through holes in the surface allows the puck to move almost without friction If the table is not accelerating, a puck placed on the table will remain at rest WW Newton’s first law WW Inertial frame of reference 114 Chapter 5 The Laws of Motion Pitfall Prevention 5.1 Newton’s First Law Newton’s first law does not say what happens for an object with zero net force, that is, multiple forces that cancel; it says what happens in the absence of external forces This subtle but important difference allows us to define force as that which causes a change in the motion The description of an object under the effect of forces that balance is covered by Newton’s second law Another statement of Newton’s first law Definition of force Pitfall Prevention 5.2 Force Is the Cause of Changes in Motion An object can have motion in the absence of forces as described in Newton’s first law Therefore, don’t interpret force as the cause of motion Force is the cause of changes in motion Definition of mass For example, an observer standing outside the train on the ground sees the puck sliding relative to the table but always moving with the same velocity with respect to the ground as the train had before it started to accelerate (because there is almost no friction to “tie” the puck and the train together) Therefore, Newton’s first law is still satisfied even though your observations as a rider on the train show an apparent acceleration relative to you A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame, and for our purposes we can consider the Earth as being such a frame The Earth is not really an inertial frame because of its orbital motion around the Sun and its rotational motion about its own axis, both of which involve centripetal accelerations These accelerations are small compared with g, however, and can often be neglected For this reason, we model the Earth as an inertial frame, along with any other frame attached to it Let us assume we are observing an object from an inertial reference frame (We will return to observations made in noninertial reference frames in Section 6.3.) Before about 1600, scientists believed that the natural state of matter was the state of rest Observations showed that moving objects eventually stopped moving Galileo was the first to take a different approach to motion and the natural state of matter He devised thought experiments and concluded that it is not the nature of an object to stop once set in motion: rather, it is its nature to resist changes in its motion In his words, “Any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of retardation are removed.” For example, a spacecraft drifting through empty space with its engine turned off will keep moving forever It would not seek a “natural state” of rest Given our discussion of observations made from inertial reference frames, we can pose a more practical statement of Newton’s first law of motion: In the absence of external forces and when viewed from an inertial reference frame, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line) In other words, when no force acts on an object, the acceleration of the object is zero From the first law, we conclude that any isolated object (one that does not interact with its environment) is either at rest or moving with constant velocity The tendency of an object to resist any attempt to change its velocity is called inertia Given the statement of the first law above, we can conclude that an object that is accelerating must be experiencing a force In turn, from the first law, we can define force as that which causes a change in motion of an object Q uick Quiz 5.1 Which of the following statements is correct? (a) It is possible for an object to have motion in the absence of forces on the object (b) It is possible to have forces on an object in the absence of motion of the object (c) Neither statement (a) nor statement (b) is correct (d) Both statements (a) and (b) are correct 5.3 Mass Imagine playing catch with either a basketball or a bowling ball Which ball is more likely to keep moving when you try to catch it? Which ball requires more effort to throw it? The bowling ball requires more effort In the language of physics, we say that the bowling ball is more resistant to changes in its velocity than the basketball How can we quantify this concept? Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity, and as we learned in Section 1.1, the SI unit of mass is the kilogram Experiments show that the greater the mass of an object, the less that object accelerates under the action of a given applied force To describe mass quantitatively, we conduct experiments in which we compare the accelerations a given force produces on different objects Suppose a force act- [...]... curved path Finally, both the magnitude and the direction of the velocity vector may change simultaneously Q uick Quiz 4.1 Consider the following controls in an automobile in motion: gas pedal, brake, steering wheel What are the controls in this list that cause an acceleration of the car? (a) all three controls (b) the gas pedal and the brake (c) only the brake (d) only the gas pedal (e) only the steering... follows these directions correctly ends up 5.00 m from the starting point 45 Review You are standing on the ground at the origin AMT of a coordinate system An airplane flies over you with constant velocity parallel to the x axis and at a fixed height of 7.60 3 103 m At time t 5 0, the airplane is directlySabove you so that the vector leading from you to it is P 0 5 7.60 3 103 j^ m At t 5 30.0 s, the position... in the figure Notice from Figure 4.5a that S vf is generally not along the direction of either S vi or S a because the relationship between these quantities is a vector expression For the same reason, from Figure 4.5b we see that S r f is generally not along the direction of S S r i, vi, or S a Finally, notice that S vf and S r f are generally not in the same direction y y ayt S vf vyf vyi S at 1 a... toward tree B, but to cover only one-half the distance between A and B Then move toward tree C, covering one-third the distance between your current location and C Next move toward tree D, covering one-fourth the distance between where you are and D Finally move toward tree E, covering one-fifth the distance between you and E, stop, and dig (a) Assume you have correctly determined the order in which... constant in this discussion, its components ax and ay also are constants Therefore, we can model the particle as a particle under constant acceleration independently in each of the two directions and apply the equations of kinematics separately to the x and y components of the velocity vector Substituting, from Equation 2.13, vxf 5 vxi 1 axt and vyf 5 vyi 1 ayt into Equation 4.7 to determine the final... depends only on the initial and final position vectors and not on the path taken As with onedimensional motion, we conclude that if a particle starts its motion at some point and returns to this point via any path, its average velocity is zero for this trip because its displacement is zero Consider again our basketball players on the court in Figure 2.2 (page 23) We previously considered only their... categorize this problem as one involving a particle moving in two dimensions Because the particle only has an x component of acceleration, we model it as a particle under constant acceleration in the x direction and a particle under constant velocity in the y direction Analyze To begin the mathematical analysis, we set vxi 5 20 m/s, vyi 5 215 m/s, ax 5 4.0 m/s2, and ay 5 0 Use Equation 4.8 for the velocity... from which they were launched Two points in this motion are especially interesting to analyze: the peak point A, which has Cartesian coordinates (R/2, h), and the point B, which has coordinates (R, 0) The distance R is called the horizontal range of the projectile, and the distance h is its maximum height Let us find h and R mathematically in terms of vi , ui , and g A ui B O x R Figure 4.9 A projectile... decathlon long jump at the 2008 Beijing Olympic Games height is the same as the initial height, we further categorize this problem as satisfying the conditions for which Equations 4.12 and 4.13 can be used This approach is the most direct way to analyze this problem, although the general methods that have been described will always give the correct answer Analyze Use Equation 4.13 to find the range... actually reduce the distance Consider the extreme case: the skier is projected at 90° to the horizontal and simply goes up and comes back down at the end of the ski track! This argument suggests that there must be an optimal angle between 0° and 90° that represents a balance between making the flight time longer and the horizontal velocity component smaller Let us find this optimal angle mathematically