We began in Chapter 2. by using moving trains to model inertial reference frames. But we made an implicit assumption beyond assuming an ideal train, with no friction and a perfectly straight track. We also assumed that there was no gravity. Einstein's famous thought experiment for discussing gravity is to consider a “freely falling” reference frame, typically a falling elevator. Objects thrown horizontally in such an elevator will not be seen to fall — there is no gravity (for a little while, at least!). But even here there is an implicit assumption, namely that the elevator is small compared to the Earth.

Return to our ideal train moving to the right, but now assume there is gravity. A ball thrown straight up will, according to an observer on the train, eventually turn around and fall back down, moving along a straight line, as shown in the first drawing in Figure 13.2. Since the acceleration due to gravity at the surface of the Earth is taken to be constant, the exact motion is described by a quadratic equation in $t$. From the ground, the same effect is seen, but combined with a constant motion (linear in $t$) to the right. The motion therefore takes place along a parabola; see Figure 13.1.

Figure 13.1: Throwing a ball in a moving train, as seen from the ground.

 
Figure 13.2: Throwing a ball in a moving train, as seen from the train.

Now suppose that that train is also accelerating to the right with constant acceleration. Then a ball thrown straight up on the train still moves along the same parabolic trajectory as before (as seen from the ground), but the rear wall of the train might now catch up with it before it lands! This shows, first of all, that Newton's laws fail in a noninertial frame.

But let's analyze the situation more carefully. Try to compensate by giving the ball a horizontal component of velocity. If the ball's initial horizontal speed is chosen appropriately, the resulting trajectory could look like the “boomerang” in the second drawing in Figure 13.2. in which (as seen from the train) the ball is thrown up at an angle, and returns along the same path!

What's going on here? Acceleration and gravity produce the same kind of effect, and what the “boomerang” is telling us is that the effective force of gravity is no longer straight down, but rather at an angle. Throw the ball up at that angle, and it goes “up” and “down” in a straight line!