STATEMENT OF PROBLEM

One mathematical model for running is the Hill-Keller model, which states that the main force resisting a runner's progress is "internal" friction; if a runner's speed at time t is given by a function v(t) then this model proposes that the opposing force is of the form -k*m*v(t), where k is some constant, and m is the mass of the runner. The propulsive force that the runner generates at any time is approximated by m*p(t). The function p(t) has units of force per unit mass. Newton's law,

Force = mass * acceleration,
implies
m*v'(t) = -m*k*v(t) + m*p(t).

The left side is just mass times acceleration, while the right side is the sum of the resistive and propulsive forces on the runner. You can divide through by m to get

v'(t) = - kv(t) + p(t).

In a sprinting race like the 100 meter dash we might reasonably assume that the runner applies maximum force throughout the race, so that
p(t) = P, where P is some constant representing the runner's maximum effort. In fact, it has been estimated that for world-class sprinters P is about 11.0 meters per second per second.

1. Solve the differential equation v'(t) + k v(t) = 11 to find the runner's velocity as a function of time. Assume that the runner starts at rest. Integrate v(t) to find the runner's position as a function of time, assuming a zero starting position.

2. Here are the split times for Ben Johnson and Carl Lewis from the 100 meter final at the World Championships in Rome in 1987. The data is in the form {time, distance} pairs, with time in seconds and distance in meters.

Input := 

johnsondata = {{0,0}, {1.84, 10}, {2.86, 20}, {3.80, 30},
{4.67, 40},{5.53, 50}, {6.38, 60}, {7.23, 70}, {8.10, 80},
{8.96, 90},{9.83, 100}};

Input := 

lewisdata = {{0,0}, {1.94, 10}, {2.96, 20}, {3.91, 30}, 
{4.78, 40},{5.64, 50}, {6.50, 60}, {7.36, 70}, {8.22, 80}, 
{9.07, 90},{9.93, 100}};

Then plot your solution for the distance as a function of time. Try to adjust the constant k in your distance function, changing the constant k until you get a best fit to the data for each runner.

Plot distance versus time for each runner and compare it to the actual data; do they agree? Does this seem like a good model for sprinter's motion?

Johnson and Lewis may well have different constants. What significance can we attach, if any, to these constants?

3. Once you have a reasonable estimate for the constant k, look at the function v(t) - differentiate your distance function from (2).

How does it behave as t tends to infinity? Is there a maximum speed for the runners? What is it?