Doubling your velocity

Question

Suppose you were to run one lap around an olympic oval with a given distance $d$, over time $t$ with velocity $v_1=d/t$. For your second lap, how fast must you travel so that your average velocity across both laps is twice your velocity during the first lap?

Answer 1

To double your speed you would have to complete two laps - a distance of $2d$ - in time $t$. But you have already spent time $t$ in travelling a distance of $d$ (your first lap), So ...

Answer 2

Another way of looking at it is to say that you need to have traversed two laps in the same time it took you to traverse one.

Since you've already taken the time to go around one lap, you'd need to do the second lap in no time at all, which is impossible.

Answer 3

Note that the condition is equivalent to

$$\bar v=\frac{D}{T}=\frac{2d}{\frac{d}{v_1}+\frac{d}{v_2}}=2v_1\implies\frac{2v_1v_2}{v_1+v_2}=2v_1\implies v_2=v_1+v_2\implies v_1=0$$

The latter is equivalent to $\frac{v_2}{v_1}\to \infty$.