Answer
The speed of the police car when it catches up to the speeder is $2v$
Work Step by Step
Let's suppose that the police car catches up to the speeder after a time $t$. The distance traveled by the speeder is $vt$.
If the police car catches up to the speeder, the police car must also travel this same distance. We can find the police car's acceleration:
$\Delta x = \frac{1}{2}at^2$
$vt = \frac{1}{2}at^2$
$a = \frac{2v}{t}$
We can find the police car's final velocity:
$v_f = v_0+at$
$v_f = 0 + (\frac{2v}{t})(t)$
$v_f = 2v$
The speed of the police car when it catches up to the speeder is $2v$
