For many years now, miners have been harvesting rare metals and other exotic materials from the asteroid belt and using Jupiter to sling-shot 10 tonne canisters of materials at the Moon, where they are collected before being refined and transferred to Earth.
At the Asteroid belt, the miners use a dedicated ship with a rail gun to insert the canisters into Jupiter's sling-shot gravity tunnel (not an actual tunnel, it's just a trajectory).
Out of seemingly nowhere (as they do), an alien mothership has appeared and is orbiting Earth, showering our homeworld in crippling EMP bursts.
Our only hope lies in our intrepid asteroid miners who hatch a plan to shoot one of the larger asteroids at the mothership.
Given that the miners are generally only used to handling smaller projectiles, would they reasonably be able to manoeuvre an asteroid into Jupiter's sling-shot tunnel and knock the alien scum out of our skies and save the day? And how long would the journey take (roughly)?
* Hollywood levels of believability
* The aliens are too stupid to notice a Manhattan sized lump of rock heading right towards them