There’s a question that gets posed toward the beginning of intro physics classes to gauge the students’ understanding of acceleration. If you fire a bullet horizontally while at the same instant dropping a bullet from the same height, which hits the ground first?

The point is to think clearly about the equation describing accelerated motion. The equation is this:

The bold letters represent vectors. Lower case r is position, a is acceleration, v is velocity. The vectors with 0 subscripted represent the starting values, so v0 is the initial velocity and r0 is the initial position.

But the whole point of a vector equation is that the equation compactly describes the motion in 3-d space. If we pick coordinates such that the x-axis is parallel with the ground and the y-axis is vertical, the equations remains true if we replace r with x (or y), a with the acceleration in that x or y direction, and v0 with the initial velocity in that direction.

So consider that fired bullet and that dropped bullet. Both of them have the same acceleration in the y direction (-9.8 m/s^2, vertically downward, from gravity). Both have the same initial velocity in the y-direction (0, since the fired bullet’s initial velocity is all in the x direction). And both have the same initial position in the y-direction – call it h, the initial height. That means both bullets are obeying the same equation in the y-direction, so they both have to reach the ground at y = 0 at the same time.

When we replace r with x, the x-direction equations are different for the two bullets, since the dropped bullet has an initial velocity of 0 in the x-direction while the fired bullet has a large initial velocity from being fired. But that matters not at all to the vertical motion.

Incidentally we can use this to find the range of a horizontally fired bullet. Take the y-direction equation, set y = 0 (which is when the bullet hits the ground), solve it for t, and plug that t back into the x-direction. The x-distance attained during that time is just the range. Doing this, I get:

For a bullet at 1,000 feet per second fired at 5 feet, that’s about 550 feet. (304 m/s and 168 m)

But until relatively recently, no one had ever actually done the experiment. It’s difficult, both in terms of dropping the bullets properly and making sure the gun fires exactly horizontally. Horizontal fire is critical, because if there is an initial vertical velocity for the fired bullet, the equation will be different from the dropped bullet and they won’t hit the ground at the same time. Nonetheless there is a group of experimenters who are very good at this sort of thing, and not so long ago they actually did the experiment. They are of course the Mythbusters:

They do the experiment and the bullets land within a few milliseconds of each other. Of course things aren’t quite mathematically perfect in real life, as air resistance and other factors will serve to distort the basic acceleration equation we’re using. Nonetheless it’s a quite good approximation and in this case works out beautifully.