Projectile Motion Calculator
45°
Horizontal Range40.775 m
Maximum Height10.194 m
Time of Flight2.883 s
The Projectile Motion Calculator models an object launched into the air with no air resistance, returning the horizontal distance it travels, the peak height it reaches, and how long it stays aloft. Supply the launch speed and the angle above the horizontal, and the tool applies the standard kinematic equations with gravity fixed at 9.81 m/s². It is built for physics homework, ballistics intuition, and sports trajectory questions.
Formula
R = v²·sin(2θ)/g ; H = v²·sin²θ/(2g) ; T = 2v·sinθ/g
- v
- Initial launch velocity (m/s)
- θ
- Launch angle above the horizontal (degrees)
- g
- Gravitational acceleration, 9.81 m/s²
How it works
- Enter the initial launch velocity in metres per second and drag the angle slider to set the launch angle between 0° and 90°.
- The calculator resolves the velocity into horizontal and vertical components, then computes range from v²·sin(2θ)/g, peak height from v²·sin²θ/(2g), and flight time from 2v·sinθ/g.
- Range peaks at a 45° launch, while a near-vertical launch maximises height but lands almost where it started.
Worked example
A ball is launched at 20 m/s at a 45° angle on level ground.
- Range = 20² × sin(90°) ÷ 9.81 = 400 × 1 ÷ 9.81 = 40.775 m.
- Max height = 20² × sin²(45°) ÷ (2 × 9.81) = 400 × 0.5 ÷ 19.62 = 10.194 m.
- Time of flight = 2 × 20 × sin(45°) ÷ 9.81 = 28.284 ÷ 9.81 = 2.883 s.
Range 40.775 m, max height 10.194 m, and 2.883 s in the air.
Frequently asked questions
- Why does a 45-degree launch give the longest range?
- Range depends on sin(2θ), which is largest when 2θ equals 90°, i.e. θ equals 45°. Angles symmetric about 45°, such as 30° and 60°, produce the same range on level ground.
- Does this calculator account for air resistance?
- No. It assumes ideal projectile motion in a vacuum over flat ground. Real-world drag shortens the range and lowers the peak height, especially for light or fast objects.
- What happens to the range at a 90-degree launch?
- At 90° the projectile goes straight up and returns to the launch point, so the horizontal range is zero even though the maximum height is at its greatest for that speed.
- Can I use different launch and landing heights?
- This tool assumes the launch and landing points are at the same elevation. For uneven terrain the flight time and range differ and require the full quadratic form of the vertical motion equation.