Answer
$v = 0.80 c$
Work Step by Step
Let $L = 0.60~L_0$.
We can find the speed:
$L = L_0~\sqrt{1-\frac{v^2}{c^2}}$
$\sqrt{1-\frac{v^2}{c^2}} = \frac{L}{L_0}$
$1-\frac{v^2}{c^2} = (\frac{L}{L_0})^2$
$\frac{v^2}{c^2} = 1-(\frac{L}{L_0})^2$
$v^2 = [1-(\frac{L}{L_0})^2]~\times c^2$
$v = \sqrt{1-(\frac{L}{L_0})^2}~\times c$
$v = \sqrt{1-(\frac{0.60~L_0}{L_0})^2}~\times c$
$v = \sqrt{1-(0.60)^2}~\times c$
$v = 0.80 c$