1.) Two runners run in a straight line and their positions are given by function

Question

1.) 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.)

Please show work!!

Explanation / Answer

let f(t) = g(t) - h(t)
Let the case starts at t=0 and ends at t=T
then g(0) = h(0) since they starts at the same time
so,f(0) = 0
Also g(T) = h(T) since they finish the race in tie
so,f(T) = 0

so there exist a point t=c where f'(c) =0 accroding to Rolle's theorem

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

g'(t) and h'(t) are velocities and they are equal to time t = c
Hence proved

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