Projectile Motion Calculator

What is projectile motion?

Projectile motion is the motion of an object launched at an angle and then acted on by gravity alone. It splits into two independent parts: horizontally the velocity stays constant, vertically the object slows at the rate g, stops and falls back. The resulting path is a parabola.

Formulas

• Velocity components: vx = v₀ × cos θ, v₀y = v₀ × sin θ
• Position: x = vx × t, y = h₀ + v₀y × t − ½ × g × t²
• Maximum height: H = h₀ + v₀y² / (2 × g)
• Time of flight: T = (v₀y + √(v₀y² + 2 × g × h₀)) / g
• Range: R = vx × T

For a launch from ground level (h₀ = 0) these reduce to the familiar short forms: T = 2 × v₀y / g and R = v₀² × sin 2θ / g. In that case the longest range comes at 45 degrees, and two angles that add up to 90 degrees (30 and 60, for example) give the same range.

Example

A ball is launched from the ground at 20 m/s, 30 degrees above the horizontal, with g = 10 m/s². The vertical velocity is 20 × sin 30° = 10 m/s and the horizontal velocity is 20 × cos 30° = 17.32 m/s. The ball reaches the top after 10 / 10 = 1 second, at a height of 10² / (2 × 10) = 5 metres. The total time of flight is 2 seconds and the range is 17.32 × 2 = 34.64 metres.

Elevated and horizontal launches

If the launch point is above the ground, enter the launch height. An angle of 0 gives a horizontal throw and a negative angle a downward throw. From a height, the angle that gives the longest range is less than 45 degrees. The calculator reports that angle and the range it gives.

The drawing

The trajectory is drawn to scale: one metre has the same length on both axes, so the shape you see is a reduced copy of the real path. You can also enter a time after launch to see where the object is and how fast it is moving at that moment.

Limits

The calculation ignores air resistance. For heavy, slow objects (a shot put, a basketball) the result is close to reality. For light or fast objects (a shuttlecock, a bullet, a golf ball) the real range can be far shorter than calculated. The ground is taken as flat, and the tool cannot be used when the landing point is higher than the launch point.