Shooting Monkeys
A classic physics problem is the following: consider a monkey hanging from a tree. The monkey has been stealing your coconuts, so you decide to shoot the monkey with a cannon. You aim directly at the monkey, but the instant you fire, the monkey lets go of the tree (presumably to avoid the cannonball) and begins to fall. Surprisingly, it turns out that no matter how fast you shoot the cannonball or where you place the cannon, the cannonball will always hit the monkey (assuming neither has hit the ground yet, ignoring air resistance, etc.).
Ordinarily you might be asked to prove this result using the laws of kinematics. For example, you could calculate the time at which the cannonball will have the same x-coordinate as the monkey, and then show that the y-coordinates of the two objects will be equal at that moment, thus proving that they always cross paths.
However, there is another way to show that the monkey has no chance of survival, which doesn't require any calculations at all! Instead, imagine that there is another monkey hanging from a tree observing the entire scene, who also lets go at the same instant. What will that monkey see?
From the second monkey's point of view, the first monkey will not move at all, and the cannonball will fly in a straight line directly towards its target. Since the cannonball travels in a straight line, it's obvious that it'll always hit the monkey. Sometimes, all it takes to solve a difficult problem is a change of perspective ;)
(Technically speaking, in the second monkey's non-inertial reference frame it's as if there is a "fictitious force" that counteracts the force of gravity, causing the cannonball and first monkey to behave as if there is no gravity at all.)