Projectile Motion

The curved flight of anything thrown — and the simple secret of splitting it in two.

Throw a ball, fire an arrow, or watch water arc from a fountain, and you are seeing projectile motion — the graceful curved path of an object moving under gravity alone. It looks complicated, but it hides a beautifully simple secret: the motion can be split into two completely independent parts.

The key insight: horizontal and vertical are independent

The single most important idea in projectile motion is that the horizontal and vertical motions do not affect each other. Horizontally, with no air resistance, nothing pushes or pulls the projectile, so it travels at constant speed. Vertically, gravity pulls it down, steadily changing its vertical velocity. These two motions happen simultaneously but separately, and combining them produces the familiar curved arc called a parabola.

The famous demonstration: a bullet fired horizontally and a bullet simply dropped from the same height hit the ground at the same time. Their vertical motions are identical; the fired bullet just travels sideways as well. Gravity does not care about horizontal speed.

The role of gravity

Near the Earth's surface, gravity accelerates everything downward at about 9.81 metres per second squared, regardless of mass. This acceleration acts only on the vertical part of the motion. With each passing second a projectile's downward velocity increases by 9.81 m/s, while its horizontal velocity stays unchanged. The vertical position follows the same equations as a freely falling object.

Try it in the calculatorUse E = m*g*h and kinematics formulas to explore falling motion.

Open calculator

Launch angle and range

When you launch a projectile at an angle, its initial velocity divides between horizontal and vertical components. A steep launch sends it high but not far; a shallow launch sends it far but low. There is a sweet spot in between. On level ground, ignoring air resistance, the maximum range comes at a launch angle of 45 degrees, which balances height and distance perfectly. Interestingly, angles that add up to 90 degrees — say 30 and 60 — produce the same range.

A worked example

A ball is thrown horizontally at 15 m/s from a cliff 20 metres high. How far does it travel before landing? First find the time to fall, which depends only on the vertical motion: falling 20 metres under gravity takes about 2 seconds. During those 2 seconds the ball moves horizontally at a constant 15 m/s, covering 15 times 2, or 30 metres. Notice we solved the vertical and horizontal parts entirely separately, then combined the results.

The parabolic path

The combination of constant horizontal velocity and uniformly accelerating vertical velocity always produces a parabola — the same elegant curve whether you are tossing a basketball, watching a stream of water, or plotting the flight of a cannonball. This is why parabolas appear so often in physics and engineering, from satellite dishes to the cables of suspension bridges.

What we left out: air resistance

Real projectiles meet air resistance, which complicates the ideal picture. Air drag shortens the range, lowers the optimal launch angle below 45 degrees, and makes the descending path steeper than the ascending one. For dense, fast, or small objects over short distances, the ideal model is excellent. For a feather, a ping-pong ball, or a long-range shell, air resistance must be taken seriously. Still, the independence of horizontal and vertical motion remains the foundation on which the more detailed analysis is built.

Key takeaways

Horizontal and vertical motions are independent and can be solved separately.
Horizontal velocity stays constant; gravity changes only the vertical velocity.
A projectile and a dropped object fall for the same time from the same height.
On level ground, 45 degrees gives the maximum range (ignoring air resistance).