Problem of the Fortnight # 5

Going at top speed, Indy 500 driver X leads his rival by a steady three miles. Only two miles from the finish line, X runs out of fuel. Thereafter, X deceleration is proportional to the square of his remaining velocity and, in the next mile, his speed exactly halves. Who wins and why?

Solution by Ben Burdett:

Well, X’s acceleration is This means that and that

, so . Solving for v so that we can integrate, we get and . We can rewrite v as v also equals so This means that and that This is great! We have the right equation now. It takes driver X’s rival to get to the finish line when driver X runs out of gas. Now, if we plug in for t, and plug in whatever k is, we can get x. If x is greater than 2mi, then driver X has already crossed the finish line when his rival crosses, and thus driver X wins. But we don’t have a value for k. Fortunately, we know that so This means that and that If we plug in for v, and 1 mile for x, we get If we plug our values for t and k into , we get , which equals which when calculated equals about 2.16mi. This means that driver X wins the race since he is across the finish line when his rival gets there.

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