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 2 (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 2
,
ks2 + S 2
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
3
4.5
6
8
9
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.
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b) With the model v = aS 2 + 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 = 2
and y = 1. 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).
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