Projectile Motion Calculator
Type a launch speed and angle and find the range, maximum height or time of flight of any projectile on level ground.
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How to solve projectile motion by hand
Split the launch velocity into horizontal (v cos θ) and vertical (v sin θ) parts — they act independently. The vertical part decides how long the projectile is in the air: it decelerates under gravity, stops rising, and falls back, taking T = 2v sin(θ)/g seconds in total. The horizontal part never changes (no air resistance), so the range is just that horizontal speed multiplied by the time of flight, which simplifies to R = v²sin(2θ)/g.
Ejemplo resuelto
A ball is launched at 20 m/s at 45°.
- Range: R = 20² × sin(90°) ÷ 9.81 = 40.8 m
- Maximum height: H = (20 × sin 45°)² ÷ (2 × 9.81) = 10.2 m
- Time of flight: T = 2 × 20 × sin 45° ÷ 9.81 = 2.88 s
See every formula worked through on its own page in the physics formula list, or fire a cannon yourself — with air resistance, different angles and even a Moon setting — in the projectile motion simulator.
Preguntas frecuentes
What is the formula for projectile motion?
Range: R = v²sin(2θ)/g. Maximum height: H = v²sin²(θ)/(2g). Time of flight: T = 2vsin(θ)/g, where v is the launch speed, θ the launch angle and g is 9.81 m/s².
How do you calculate the range of a projectile?
Multiply the launch speed squared by sin(2 × angle), then divide by g (9.81 m/s²). A ball launched at 20 m/s at 45° lands 40.8 m away.
¿Qué ángulo da el máximo alcance?
45°, for a given launch speed on level ground with no air resistance — that is where sin(2θ) reaches its largest value, 1.
Does air resistance change the answer?
Yes — air resistance shortens the range and lowers the maximum height, and the effect grows with speed. This calculator ignores it, as do the standard formulas; try the simulator below with air resistance switched on to see the difference.
