Projectile motion calculator

Formula

How it works

Launching at speed v and angle θ splits into a horizontal part that never changes and a vertical part slowed by gravity. The horizontal and vertical motion are independent, and combining them gives the range, the height at the top of the path, and the total time in the air.

Worked example

The same calculation with the default values:

• Launch speed 20 m/s
• Launch angle 45 °

1. range = 20² × sin(2 × 45°) ÷ 9.80665 = 40.788649
2. maximum height = 20² × sin²(45°) ÷ (2 × 9.80665) = 10.197162
3. time of flight = 2 × 20 × sin(45°) ÷ 9.80665 = 2.884193

Result: Range: 40.78864852 m · Maximum height: 10.19716213 m · Time of flight: 2.884193 s

Common mistakes

• Forgetting that this model ignores air resistance - real projectiles (especially light or fast-moving ones) travel a shorter range than this idealized formula predicts.
• Using a launch angle in degrees directly inside the trigonometric part of the formula without converting to radians (the calculator handles this internally, but a manual check should convert first).
• Confusing maximum height and range - maximum height depends on sin²θ while range depends on sin(2θ), so the angle that maximizes one does not maximize the other (maximum range is at 45°, but maximum height keeps increasing up to 90°).