A classic in introductory Physics classes is the "shoot the monkey" demonstration. A gun is aimed at a monkey sitting high in a tree. As soon as the bullet is fired, the monkey drops from the tree. Ignoring air resistance, where should one aim the gun in order to hit the monkey as it falls?

Answer: aim directly where the monkey is sitting.

Many students find this counter-intuitive, and believe that one must "lead" the monkey or in some other manner aim below it to compensate for the fall. This isn't true, though, because the bullet falls at the same rate as the monkey.

If you were to fire the gun at the monkey with zero gravity, the bullet would travel along a straight line, and the trajectory would always points towards the monkey, because the monkey doesn't fall. Now, add gravity, which applies equally to the monkey and the bullet. While the bullet doesn't travel along a straight line with respect to the gun's reference frame, it does travel along a straight line with respect to the monkey's reference frame.

To put it yet another way, consider the bullet to instead be a guided missile pointed at a monkey in zero gravity. We give the missile an initial X velocity by firing it. As soon as the missile is fired, the monkey says "uh oh!" and takes evasive action by accelerating in the Y direction. But the missile is smart -- it sees the monkey's escape, and (instantly) starts firing its Y-axis engines to match the monkey's acceleration. The downward component of the monkey's Y-motion is always matched by the downward component of the missile's Y-motion (even when the missile's upward component is greater) and thus the monkey is blown to bits.

This explanation utilizes an accelerating (or "non-inertial") reference frame. Such reference frames are typically (perhaps too often) avoided since, to paraphrase Einstein, the laws of physics must be the same in all inertial reference frames. Keeping Newton's Second Law (F=ma) "the same" requires that one avoid accelerating reference frames (no extra accelerations or "fictitious forces" introduced due to the reference frame). However, how the laws must be modified to jump to an accelerating reference frame are well defined and sometimes very useful as illustrated by the shoot the monkey discussion.

Let us take this discussion one step farther. Suppose one is abducted by aliens and taken (sedated) into deep space, far from any gravitational force sources. If the space ship is at rest, or in constant velocity motion, you (and all inside) would feel weightless. Now suppose the aliens want to wake you up and observe you while fooled into thinking you are still on earth. They carefully set up a shoot the monkey demonstration. At the instant just after the bullet leaves the gun and the "monkey" lets go of its tree the aliens fire their rocket motors to impart precisely a=+9.8m/s2 and wake you up instantly. You feel your weight (-mg) you see the monkey fall with g=-9.8m/s2, you see the bullet follow a parabolic trajectory (consistent with g=-9.8m/s2) and strike the "monkey" just as in physics class.

Of course, an observer outside the space craft sees the monkey remain motionless (he let go of the branch before the rocket fired), and the bullet travel in a straight line (it had cleared the muzzle just before the rocket fired ) to hit the monkey. The observer sees you, and the floor on which you rest, accelerating at a=+9.8m/s2 upward toward the monkey and the bullet.

This is a classic example of Einstein's Equivalence Principle from which he formed his theory of General Relativity (the theory of gravity in terms of the curvature of space). The Equivalence Principle basically says that, experiments in the presence of mass and a gravitational interaction must be equivalent to those carried out with no gravity, but in a reference frame having the precise acceleration needed to simulate the gravitational field. In this formulation the path of light, the quanta of which (photons) have no mass, also follows a curved path in a gravitational field as they must in an accelerating reference frame. .

These are my lecture notes. Please feel free to use them elsewhere with attribution.