Average speed question of a car on an oval track

Question

On an oval track, a car averages $30$ M.P.H. (miles per hour) for the first lap (a lap is $1$ mile around the track). At what average speed must that same car drive the 2nd lap on that same track so that the average speed of both laps is $60$ M.P.H.?

Answer 1

Average speed = Distance / Time = harmonic mean of two speeds

60 = (2*30*x)/(30+x); 30+x=x; 0=30

This is not true, so there is no solution.

I'm not sure if this is correct, so if you want to check my work...

Answer 2

First lap lasts $1/30$ hour, let $v_2$ M.P.H. the speed for the second lap, a time $t_2=\frac{1}{v_2}$ hours is required, then the average speed of the two laps is $$\overline{v}=\frac{\overbrace{1}^\text{Distance covered in the first lap}+\overbrace{1}^\text{Distance covered in the second lap}}{\underbrace{\frac{1}{30}}_\text{Time needed for the first lap}+\underbrace{\frac{1}{v_2}}_\text{Time needed for the second lap}}=\frac{60v_2}{v_2+30}$$ And $\overline{v}=60$ M.P.H. iff \begin{align*} &&\frac{60v_2}{v_2+30}&=60\\ \iff &&60v_2&=60v_2+1800\\ \iff &&0&=1800 \end{align*} Therefore, it's not possible to attain an average speed of $60$ M.P.H.

Answer 3

A friend asked me this question (as a brain teaser I suppose), and I got it right the first time (although my thinking came across some wrong solutions first which I quickly fixed before emailing him my correct answer). Think of it like this. Imagine $2$ cars on this oval track, call them cars A and B. A drives the first lap (mile) at $30$ MPH average speed, therefore taking $2$ minutes to drive the first $1$ mile. Since the question is asking for an average speed of $60$ MPH over $2$ laps ($2$ miles), we know that it takes $2$ minutes to drive $2$ miles at $60$ MPH. Therefore there is no speed (not even the speed of light) for the 2nd lap of car A that will give it an average speed of $60$ MPH for the $2$ miles because after $2$ minutes, car A is only at the $1$ mile marker and has $1$ more mile to go!

It may help if you imagine car B going a constant $60$ MPH, after $2$ minutes car B would be done the "race" but car A would only be halfway, so even if it had nitrous, a supercharger, dual turbochargers... there is no way it will ever get the $60$ MPH average speed for the $2$ miles with that slow start.

Moral of the story... don't be a "pussyfooter" when it comes to racing cars.

Now had I changed the problem to $40$ MPH for the first lap, then there WOULD be a solution.