Projectile Motion Explained
When you throw a ball, kick a soccer ball into the air, or fire a cannon, the resulting path is one of the most recognizable shapes in physics: a parabola. Projectile motion analyzes this by splitting the object's motion into two completely independent pieces — a constant horizontal velocity, and a vertical velocity that gravity steadily accelerates downward. Neither piece knows about the other; treating them separately, then recombining the results, is what makes it possible to predict exactly how far a projectile will travel, how long it stays in the air, and how high it climbs, using nothing more than basic kinematics.
The parabolic path
Combining constant vx with accelerating vy traces a parabola. Launch angle and initial speed determine the maximum height and horizontal range.
Free Fall & Projectile Motion
Drop anything — a coin or a bowling ball — and gravity speeds it up at exactly the same rate, so they hit the ground together. Toss something sideways instead, and it still falls straight down just as fast, because its sideways motion and its falling motion don't affect each other at all.
The formula
h = ½ · g · t²
- h — height fallen (m): how far the object has fallen from its starting point
- g — gravitational acceleration (m/s²): how quickly falling speed increases each second — the same for every object
- t — time (s): how long the object has been falling
Worked example
A ball is dropped from rest and falls for 2 seconds. How far has it fallen? (use g = 9.8 meters per second squared)
- t = 2 s
- g = 9.8 m/s²
- h = ½ · g · t²
- h = 0.5 × 9.8 m/s² × (2 s)²
h = 19.6 m
Test yourself
The same ball is dropped again, but this time it falls for 3 seconds instead of 2. How far does it fall?
- Correct answer: 44.1 m
- 88.2 m
- 14.7 m
Right! h = ½ × 9.8 m/s² × (3 s)² = 44.1 meters — squaring the time matters a lot.
One ball is dropped straight down off a table. At the same instant, a second ball is launched horizontally off the same table. Which one hits the floor first?
- The dropped ball
- The horizontally launched ball
- Correct answer: Both hit at the same time
Exactly — sideways speed doesn't change the falling motion, so both balls hit the floor at the same time.
Where you see this
A pitcher's fastball drops as it crosses the plate, a diver's fall speeds up right up to the splash, and a Nerf dart lobbed sideways still lands at the same time as one just dropped straight down. Any time something moves through the air near Earth's surface with no engine of its own, this is the motion underneath.
Common mistakes
It's tempting to think a heavier object falls faster — in the absence of air resistance, mass doesn't affect the fall rate at all, only air drag can slow a lighter or less aerodynamic object down. It's also easy to assume a projectile launched sideways falls slower than one just dropped, but the two hit the ground at the same instant: horizontal motion and vertical fall are entirely independent.
How it connects
This is usually the first motion students meet with acceleration that isn't caused by a push or pull they can see — gravity does the accelerating. Newton's second law, next, explains why every falling object accelerates at the same rate regardless of mass: F = ma means a heavier object needs proportionally more force to accelerate it, and gravity supplies exactly that much more.