Two runners run in a straight line and their positions are given by functions g(

Question

Two runners run in a straight line and their positions are given by functions g(t) and h(f), where t is the time in seconds, and g(t) is the number of meters from the starting point for the first runner and h(t) is the number of meters from the starting point for the second runner. Assume that g(t) and h(t) are differentiable functions. Suppose that the runners begin a race at the same moment and end the race in a tie. Carefully explain why at some moment during the race they have the same velocity. (Suggestion: Consider the function f(t) = g(t) - h(t) and use Rolle's Theorem.)

Explanation / Answer

velocity is the derivative of the distance covered (or the position) with respect to the time t
=> v1(t) = (d/dt) g(t)
=> v2(t) = (d/dt) h(t)
Now v1(t) = v2(t) if (d/dt) f(t) = 0 .
Now to be able to apply Rolle we need to have f(0) = f(T) = 0, with T the race time.
f(T) = 0 as they end in a tie.
But the assignment is a bit confusing as it is not explicitly told that they start at the same starting point.
To have f(0) = g(0) - h(0) = 0 , we need g(0) = h(0) so they do start at the same place.
Then we apply Rolle, and Rolle says that there exists a t in [0, T] for which f '(t) = 0 => v1(t) = v2(t) .

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