World Class Sprints - A First-Order Linear Model

David Smith

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.

  1. What is an appropriate initial condition for the model?

  2. 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.]

  3. What is the limiting behavior of v as t becomes large? What meanings can you attach to the parameters A and b?