World Class
Sprints
Part 1: A First-Order
Linear Model
In 1973, J. B. Keller published
a theory of competitive running, in which he proposed that the speed of
a sprinter (up to 300 meters) could be modeled by the differential equation
dv/dt = A - v/b,
where v(t) is the
speed at time t. At the time of Keller's work, reasonable values
for the constants were A = 12.2 m/sec2 and b = 0.892 sec.
- What is an appropriate
initial condition for the model?
- Solve the initial value
problem symbolically, using the parameters A and b -- don't
substitute numbers yet. You should be able to do this step with pencil
and paper, but use your helper application if you need to. [You may have
encountered problems of this type in many different contexts: RL circuits,
Newton's Law of Cooling, exponential growth and decay, velocity in a resisting
medium, and mixing problems.]
- What is the limiting behavior
of v as t becomes large? What meanings can you attach to
the parameters A and b?
