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.
Pick the ○ next to the value you want to find, fill in the others, and it solves as you type.
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.
Worked example
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.
الأسئلة الشائعة
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.
What angle gives the maximum range?
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.
