Answer
The catapult can be used to move the asteroid closer to Earth.
Work Step by Step
Consider what happens when the catapult throws stone into space.
Before the catapult starts to throw, we assume the asteriod, the catapult and the stones are static; therefore, there is no momentum in the system, $\sum p_0=0$
When the catapult throws a stone, we assume the mass of the asteroid and the catapult is $M$ and these two have velocity $V$, while the stone has mass $m$ and velocity $v$. The total momentum is conserved, so $$MV+mv=\sum p_0=0$$ $$mv=-MV$$
This equation means as the catapult throws a stone, the asteroid and the catapult recoil in the opposite direction. So if the throwing direction is appropriately located, the catapult can be used to move the asteroid closer to Earth.
