Projectile Motion: Two Completely Separate Falls Happening at Once

Around 1609, Galileo Galilei started chasing down exactly this question — using cannonballs instead of basketballs — and by the time he published his full results in 1638, he'd worked out something genuinely elegant: a projectile's motion is really two completely separate motions happening at once. Horizontally, it coasts along at a constant speed, like nothing is happening at all. Vertically, it's in ordinary free fall. Neither motion affects the other, and combining a constant sideways drift with a downward acceleration produces a specific curve — a parabola — every single time. The same parabola shows up in a thrown football, a fired bullet, and, eventually, a satellite orbit.

When a basketball leaves a shooter's hands, it moves forward and falls toward the ground at the same time. It's natural to assume those two things must interact somehow — that moving fast forward should change how quickly it falls. They don't. That surprising fact is called the independence principle, and it's the key that makes projectile motion far simpler than it looks.

Here's a demo worth picturing: roll a ball off the edge of a table so it launches horizontally, at the exact same instant a second, identical ball is simply dropped straight down from the same height. Which one hits the floor first? Neither — they land at the exact same moment, every time. The rolled ball travels much farther horizontally before landing, but its vertical fall — how long it takes to reach the ground — is identical to the dropped ball's. Horizontal speed has zero effect on fall time.

The reason is that horizontal and vertical motion are governed by completely different rules and don't talk to each other. Horizontally, there's no acceleration acting (ignoring air resistance), so horizontal velocity simply stays constant for the whole flight — distance traveled is just velocity times time, $x = v_xt$. Vertically, the object is in ordinary free fall, exactly like Topic 1.3, Session 6 (Free Fall), completely unaffected by how fast it's moving sideways: $\Delta y = \tfrac{1}{2}gt^2$.

That independence gives a clean strategy for solving any horizontally-launched projectile problem: solve the vertical motion first to find out how long the object is in the air (since vertical motion doesn't need to know anything about horizontal speed), then plug that time into the horizontal equation to find how far it traveled. Vertical first, horizontal second — always in that order, because the vertical fall is the only thing that determines when the object reaches the ground, and everything else follows from there.

Try It: Cannon Command

Aim a cannon and predict where the projectile lands: Kinematics Castle: Cannon Command.

Try It: Stunt Truck Jump

Choose the ramp angle and speed to jump a gap. Kinematics Castle: Stunt Truck Jump

Videos

• Introduction to Projectile Motion — Flipping Physics

Practice Problems

1. A ball rolls off a table 4.9 m high at a horizontal velocity of 3 m/s. Find the time to hit the floor and the horizontal distance traveled.
2. A marble rolls off a ledge 19.6 m high at a horizontal velocity of 5 m/s. Find the time to fall and the horizontal distance traveled.
3. A stone rolls off a cliff 44.1 m high at a horizontal velocity of 10 m/s. Find the time to fall and the horizontal distance traveled.
4. A ball launched horizontally from a height of 19.6 m lands 24 m away. Find the horizontal launch velocity.
5. A stone rolls off a cliff 78.4 m high at 15 m/s. Find (a) the vertical velocity component at impact, (b) the resultant speed at impact (combining horizontal and vertical components), and (c) the angle below the horizontal at which it strikes the ground.
6. A dart is thrown horizontally at 4 m/s from a height of 1.225 m, aimed at a target whose bullseye is exactly at the throw height (1.225 m off the ground), positioned 2 m away horizontally. a) Find the time for the dart to travel the 2 m to the target. b) Find how far the dart has fallen vertically in that time. Based on this, does the dart hit the bullseye, or does it land somewhere else? Explain. c) To hit the bullseye instead, should the thrower increase or decrease the horizontal velocity? Explain your reasoning conceptually (no need to recalculate).
7. A classmate claims: "A bullet fired perfectly horizontally from a gun will take longer to hit the ground than an identical bullet simply dropped from the same height, because the fired bullet has to travel much farther overall." Evaluate this claim — is it true or false? Justify your answer.