Two racecars start a race at the same place and finish the race in a perfect tie. Provethere was at least one moment during the race when both cars were going at exactlythe same speed. (HINT: Let f(t) represent the location of the first car at time t, andlet g(t) represent the location of the second car at time t. Let h(t) = f(t) – g(t). Ithink the function h(t) merits some serious consideration.)%3|

Answered: Two racecars start a race at the same…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Two racecars start a race at the same place and finish the race in a perfect tie. Prove
there was at least one moment during the race when both cars were going at exactly
the same speed. (HINT: Let f(t) represent the location of the first car at time t, and
let g(t) represent the location of the second car at time t. Let h(t) = f(t) – g(t). I
think the function h(t) merits some serious consideration.)
%3|

Expert Solution

Step 1

The objective here is to show that the speed of both cars were same at least in one moment during the given race.

Given: Both racers start at a same place and finish race in a perfect tie.

Step 2

Consider that f(t), and g(t) are the position of first and second car at any time t, then their speed respectively will be f'(t) and g'(t).

Now consider h(t)=f(t)-g(t), and let t₀ and t₁was start time and finish time of race by both racers, note that from the given information that both the racers finishes and started race at same time and their positions are same at both the times, i.e.

f(t₀)=g(t₀), and f(t₁)=g(t₁)

That implied h(t₀)=f(t₀)-g(t₀)=0, and h(t₁)=f(t₁)-g(t₁)=0

Thus,

h(t₀)=h(t₁)

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