Projectile Motion Calculator
Find the range, peak height and flight time of a launched projectile.
Please note: Results use idealised models — point masses, no air resistance, ideal gases and uniform materials. Real experiments deviate. Use these for study and estimation, not for engineering sign-off.
What the Projectile Motion Calculator does
A projectile’s motion splits cleanly into two independent parts: constant horizontal velocity, and vertical motion under gravity alone. Solving the vertical equation for when the object returns to ground level gives the flight time, and multiplying that by the horizontal velocity gives the range.
Formula
vₓ = v₀ cos(θ), v_y = v₀ sin(θ)Flight time t = (v_y + √(v_y² + 2gh₀)) ÷ gRange R = vₓ × tPeak height H = h₀ + v_y² ÷ (2g)
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Launch speed | number | Yes | — |
| Launch speed unit | one of 5 options | Yes | — |
| Launch angle | ° | Yes | Measured from the horizontal. Accepts more than -90, up to 90. |
| Launch height | number | Optional | Height above the landing surface. |
| Launch height unit | one of 9 options | Yes | — |
| Gravitational acceleration | m/s² | Optional | Standard gravity on Earth is 9.80665 m/s². The Moon is 1.62, Mars 3.72. Accepts 0 or more. |
How to use it
- Choose Launch speed unit and Launch height unit.
- Enter Launch speed and Launch angle.
- Optionally add Launch height and Gravitational acceleration.
- Select Calculate.
Worked example
A ball thrown at 20 m/s at 45° from ground level.
- Speed
- 20 m/s
- Angle
- 45°
- Height
- 0 m
vₓ = v_y = 14.142 m/s. Flight time 2.884 s, range 40.79 m, peak height 10.20 m.
Frequently asked questions
Why is 45° the best angle?
It balances hang time against horizontal speed. Steeper gives more time but less forward velocity; shallower gives more velocity but less time. From a raised launch point the optimum drops below 45°.
How much does air resistance change this?
For a thrown ball, the real range is typically 5–15% shorter. For fast, light objects like a shuttlecock or a golf ball the model breaks down entirely.