Bài giảng vật lý đại cương chương 2

Chapter Motion in 1-D 2.0 Some mathematical concepts 2.1 Position, Velocity and Speed 2.2 Instantaneous Velocity and Speed 2.3 Acceleration 2.4 One-Dimensional Motion with Constant Acceleration 2.5 Freely falling Object 2.6 Kinematic Equations Derived from Calculus Defining a Coordinate System One-dimensional coordinate system consists of: • a point of reference known as the origin (or zero point), • a line that passes through the chosen origin called a coordinate axis, axis one direction along the coordinate axis, chosen as positive and the other direction as negative, and the units we use to measure a quantity Scalars and Vectors • A scalar quantity is one that can be described with a single number (including any units) giving its magnitude magnitude • A Vector must be described with both magnitude and direction direction A vector can be represented by an arrow: •The length of the arrow represents the magnitude (always positive) of the vector •The direction of the arrow represents the direction of the vector A component of a vector along an axis (one-dimension) A UNIT VECTOR FOR A COORDINATE AXIS is a dimensionless vector that points in the direction along a coordinate axis that is chosen to be positive A one-dimensional vector can be constructed by: •Multiply the unit vector by the magnitude of the vector •Multiply a sign: a positive sign if the vector points to the same direction of the unit vector; a negative sign if the vector points to the opposite direction of the unit vector A component of a vector along an axis=sign × magnitude Difference between vectors and scalars • The fundamental distinction between scalars and vectors is the characteristic of direction direction Vectors have it, and scalars not • Negative value of a scalar means how much it below zero; negative component of a vector means the direction of the vector points to a negative direction Check Your Understanding Which of the following statements, if any, involves a vector? (a) I walked km along the beach (b) I walked km due north along the beach (c) I jumped off a cliff and hit the water traveling at 25 km per hour (d) I jumped off a cliff and hit the water traveling straight down at 25 km per hour (e) My bank account shows a negative balance of –25 dollars 2.1 Position, Velocity and Speed • The world, and everything in it, moves • Kinematics: describes motion • Dynamics: deals with the causes of motion One-dimensional position vector • The magnitude of the position vector is a scalar that denotes the distance between the object and the origin • The direction of the position vector is positive when the object is located to the positive side of axis from the origin and negative when the object is located to the negative side of axis from the origin Displacement • DISPLACEMENT is defined as the change of an object's position that occurs during a period of time • The displacement is a vector that points from an object’s initial position to its final position and has a magnitude that equals the shortest distance between the two positions • SI Unit of Displacement: meter (m) Example 2: Determine the displacement in the following cases: (a) A particle moves along a line from to (b) A particle moves from to (c) A particle starts at m, moves to m, and then returns to m 2.3 Acceleration Change in velocity Average acceleration= Elapsed time r r r r v2 − v1 ∆v a = = t2 − t1 ∆t SI Unit of Average Acceleration: meter per second squared (m/s2) Instantaneous acceleration: r r 2r d x r dv d dx a= = ( )= dt dt dt dt Acceleration is the 2nd derivative of position with respect to time • An object is accelerated even only direction changes (e.g uniformly circular motion, next chapter) • It is important to realize that speeding up is not always associated with an acceleration that is positive Likewise, slowing down is not always associated with an acceleration that is negative The relative directions of an object's velocity and acceleration determine whether the object will speed up or slow down EXERCISE A cat moves along an x axis What is the sign of its acceleration if it is moving (a) in the positive direction with increasing speed, (b) in the positive direction with decreasing speed, (c) in the negative direction with increasing speed, and (d) in the negative direction with decreasing speed? EXAMPLE 7: Position and Motion A particle's position on the x axis is given by with x in meters and t in seconds • (a) Find the particle's velocity function and acceleration function • (b) Is there ever a time when vx =for ? • (c) Describe the particle's motion t≥0 2.4 Motion diagram 2.5 Motion with Constant Acceleration 2.6 Free-fall Acceleration 2.7 Kinematic equations Free-Fall Acceleration • 1564 - 1642 • Applied scientific method • Galileo formulated the laws that govern the motion of objects in free fall • Also looked at: – – – – Inclined planes Relative motion Thermometers Pendulum Equations of Motion with Constant Acceleration v2 x = v1x + ax ∆t ∆x = (v1x + v2 x )∆t 2 ∆x = v1x ∆t + ax ∆t 2 v2 x = v1x + 2ax ∆x Example A Falling Stone A stone is dropped from rest from the top of a tall building After 3.00 s of freefall, (a)what is the velocity of the stone? (b)what is the displacement y of the stone? Example An Accelerating Spacecraft The spacecraft shown in the figure beside is traveling with a velocity of +3250 m/s Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2 What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing Example 10 Spotting a police car, you brake your Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration • What is that acceleration? • (b) How much time is required for the given decrease in speed? Graphical I n Integration t e g r a t i o n i n M o t i o n in Motion Analysis Conceptual Question A honeybee leaves the hive and travels km before returning Is the displacement for the trip the same as the distance traveled? If not, why not? Two buses depart from Chicago, one going to New York and one to San Francisco Each bus travels at a speed of 30 m/s Do they have equal velocities? Explain One of the following statements is incorrect (a) The car traveled around the track at a constant velocity (b) The car traveled around the track at a constant speed Which statement is incorrect and why? At a given instant of time, a car and a truck are traveling side by side in adjacent lanes of a highway The car has a greater velocity than the truck Does the car necessarily have a greater acceleration? Explain The average velocity for a trip has a positive value Is it possible for the instantaneous velocity at any point during the trip to have a negative value? Justify your answer An object moving with a constant acceleration can certainly slow down But can an object ever come to a permanent halt if its acceleration truly remains constant? Explain [...]... = (3 m/s)t – (2 m); (2) x = (–4 m/s2)t2 – (2 m); (3) x = (–4 m/s2)t2; (4) x = 2 m • (a) In which situations is the velocity of the particle constant? • (b) In which is the vector pointing in the negative x direction? 2. 3 Acceleration Change in velocity Average acceleration= Elapsed time r r r r v2 − v1 ∆v a = = t2 − t1 ∆t SI Unit of Average Acceleration: meter per second squared (m/s2) Instantaneous... ∆t 1 ∆x = (v1x + v2 x )∆t 2 1 2 ∆x = v1x ∆t + ax ∆t 2 2 2 v2 x = v1x + 2ax ∆x Example 8 A Falling Stone A stone is dropped from rest from the top of a tall building After 3.00 s of freefall, (a)what is the velocity of the stone? (b)what is the displacement y of the stone? Example 9 An Accelerating Spacecraft The spacecraft shown in the figure beside is traveling with a velocity of + 325 0 m/s Suddenly... motion t≥0 2. 4 Motion diagram 2. 5 Motion with Constant Acceleration 2. 6 Free-fall Acceleration 2. 7 Kinematic equations Free-Fall Acceleration • 1564 - 16 42 • Applied scientific method • Galileo formulated the laws that govern the motion of objects in free fall • Also looked at: – – – – Inclined planes Relative motion Thermometers Pendulum Equations of Motion with Constant Acceleration v2 x = v1x +... (pair 2) –3 m, –7 m; (pair 3) 7 m, –3 m • (a) Which pairs give a negative displacement? • (b) Calculate the value of the displacement in each case using vector notation Velocity and Speed A student standing still at a horizontal distance of 2. 00 m to the left of a spot of the sidewalk designated as the origin A student is walking slowly Her horizontal position starts at a horizontal distance of 2. 47... velocity Average acceleration= Elapsed time r r r r v2 − v1 ∆v a = = t2 − t1 ∆t SI Unit of Average Acceleration: meter per second squared (m/s2) Instantaneous acceleration: r r 2r d x r dv d dx a= = ( )= 2 dt dt dt dt Acceleration is the 2nd derivative of position with respect to time • An object is accelerated even only direction changes (e.g uniformly circular motion, next chapter) • It is important to... Suppose that to pump the gasoline, pay for it, and walk back to the truck takes you another 45 min What is your average speed from the beginning of your drive to your return to the truck with the gasoline? 2. 2 Instantaneous Velocity and Speed The instantaneous velocity is the derivative of the object’s position with respect to time r r ∆x dx dx ) r v = lim = = i dt dt ∆t →0 ∆t • The instantaneous velocity... traveling with a velocity of + 325 0 m/s Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2 What is the velocity of the spacecraft when the displacement of the craft is +21 5 km, relative to the point where the retrorockets began firing Example 10 Spotting a police car, you brake your Porsche from a speed of 100 km/h to a speed of... speed? Graphical I n Integration t e g r a t i o n i n M o t i o n in Motion Analysis Conceptual Question 1 A honeybee leaves the hive and travels 2 km before returning Is the displacement for the trip the same as the distance traveled? If not, why not? 2 Two buses depart from Chicago, one going to New York and one to San Francisco Each bus travels at a speed of 30 m/s Do they have equal velocities?... velocity, and what is his average speed? EXAMPLE 6 You drive a truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops Over the next 30 min, you walk another 2. 0 km farther along the road to a gasoline station • (a) What is your overall displacement from the beginning of your drive to your arrival at the station? • (b) What is the time interval from the beginning... these data, determine the average velocity for each run • Example 5: find the average velocity for the student motion represented by the graph shown in the figure below between the times t1 = 1.0 s and t2 = 1.5 s Average Speed Average speed is defined as: total distance v= ∆t Check Your Understanding A straight track is 1600 m in length A runner begins at the starting line, runs due east for the full