Answer
The correct answer is: (a) $\sqrt{3}~v$
Work Step by Step
We can find an expression for the speed when the catapult is stretched a distance $d$:
$\frac{1}{2}mv^2 = \frac{1}{2}kd^2$
$v = d~\sqrt{\frac{k}{m}}$
We can find an expression for the speed when the catapult is stretched a distance $d$ and the mass of the pebble is $\frac{m}{3}$:
$\frac{1}{2}(\frac{m}{3})v_2^2 = \frac{1}{2}kd^2$
$v_2 = d~\sqrt{\frac{3k}{m}}$
$v_2 = \sqrt{3}\times d~\sqrt{\frac{k}{m}}$
$v_2 = \sqrt{3}~v$
The correct answer is: (a) $\sqrt{3}~v$
