BTL-Chude-2-2

Applied Mathematics Department – o O o – Teamwork Project Problem 1) A bungee jumper jumps from a mountain with the downward vertical velocity v described by the mathematical model dv dt = g − cd m v2[.]

Applied Mathematics Department Teamwork Project –oOo– Problem 1) A bungee jumper jumps from a mountain with the downward vertical velocity v described by the mathematical model: dv dt = g − cmd v (see the picture), where m is the mass of jumper and cd is called drag coefficient a) Suppose that the jumper is initial at rest, find analytically the expression of v b) Let g = 9.8(m/s2 ), m = 68.1(kg), cd = 0.25(kg/m) and the jumper is initial at rest, establish the table to compute the velocity of the jumper for the first 10 seconds with step size h = 1(s) by using modified Euler’s and Runge-Kutta’s method Compare the results to the exact values found in a) c) Using the result of a) and the bisection method, the secant method to determine the drag coefficient for a jumper with the weight of 95(kg) and the velocity v = 46(m/s) after 10 seconds of fall until the relative error is less than 5%(Guess the isolated interval containing root) Problem 2) Enzymes act as catalysts to speed up the rate of chemical reactions in living cells In most cases, they convert one chemical, the substrate, into another, the product The Michaelis-Menten equation is commonly used to describe such reactions: v= vm S , ks2 + S where v = the initial reaction velocity, vm = the maximum initial reaction velocity, S = substrate concentration, and ks = a half-saturation constant a) The relationship between S and v is provided in the following table: S 1.3 1.8 4.5 v 0.07 0.13 0.22 0.275 0.335 0.35 0.36 Using the least square method to determine vm and ks by converting the given model into linear model b) With the model v = aS + bS + c, again, using the least square method to determine a, b, c Can you estimate that which model(linear or parabola) gives the better approximation? Problem 3) In biology, the predator-prey model is used to observe the species interaction One model is proposed by Lotka-Volterra as: dx = ax − bxy dt dy = −cy + dxy dt where x, y are the number of prey and predator,respectively, a = the prey growth rate,c = the predator death rate, b and d = the rates characterizing the effect of the predator prey interactions on the prey death and the predator growth, respectively t is time measured in month a) Given the following data: a = 1.2, b = 0.6, c = 0.8, d = 0.3 with initial conditions of x = and y = Find the number of prey and predators after 10 months with modified Euler method with step size h = 0.625 b) With the found data, construct the natural cubic spline for x and y Plot in one figure the graphs of x(t), y(t)