Answer: $\large\frac{1}{3}$

A ball 1 moving with velocity u collides with an identical ball 2 at rest, then after collision

$v_2=\bigg(\large\frac{1+e}{2}\bigg)$$u$ and $v_1=\bigg(\large\frac{1-e}{2}\bigg)$$u$, where $u$ is initial speed of ball 1

Given: $ v_2=2 v_1$, solving for $e$, we get:

$v_2=\bigg(\large\frac{1+e}{2}\bigg)$$u = 2v_1=2\bigg(\large\frac{1-e}{2}\bigg)$$u$

$e=\large\frac{1}{3}$