Two runners run in a straight line and their positions are g

Two runners run in a straight line and their positions are given by functions g(t) and h(t), 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 dining the race they have the same velocity. (Suggestion: Consider the function f(t) = g(t) - h(t) and use Rolle\'s Theorem.)

Solution

According to Rolle\'s Theorem : any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero.

In this case f(t) would be differentiable as both g(t) and h(t) are differentiable functions as given in the question.And f(t) attains the value 0 at two distinct points : one at the start of the race and another at the end as they start and finish the race at the same time.So applying Rolle\'s Theorem,f(t) must have a point between the start and finish time where the first derivative of f(t) would be zero.And the first derivative is nothing but the relative velocity of the first runner with respect to the second runner.Zero relative velocity implies they must have the same velocity at that moment in the race.