(5 points) Two runners run in a straight line and their positions are given by f

Question

(5 points) 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 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

Let the race started at t=0 and ended at t=T

let f(t) = g(t) - h(t)

since g(t) and h(t) are continous in [0,T], f(t) will also be continuous in [0,T]
since g(t) and h(t) are differentiable in (0,T), f(t) will also be differentiable in (0,T)

since h(0) = g(0) and h(T) = g(T)

we can say that f(0) = 0 and f(T) = 0

som f(t) satisfies all the requirement for Rolles theorem

So, according to rolles theorem, there must be a value c such that f'(c) = 0

f(t) = g(t) - h(t)
f'(t) = g'(t) - h'(t)
put t=c
f'(c) = g'(c) - h'(c) = 0
g'(c) - h'(c) = 0
g'(c) = h'(c)

so, velocity is equal at atleast one point

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