Quick answer
This projectile motion calculator finds the range, maximum height and time of flight of anything launched through the air. Enter the launch speed, the angle above the horizontal and, if needed, the launch height. Air resistance is ignored, as in most textbook problems.
Projectile range, height and time of flight
How to Use the Projectile Motion Calculator
- Enter the launch speed v₀ in m/s.
- Enter the launch angle θ in degrees, measured up from the horizontal (0° to 90°).
- Enter the launch height above the landing point, or leave it at 0 for level ground.
- Keep g = 9.81 m/s² for Earth, or change it for another planet (for example 1.62 on the Moon).
- Press Calculate to see every result and the working.
Projectile Motion Formulas
The calculator treats horizontal and vertical motion separately. Horizontally there is no acceleration; vertically, gravity pulls down at g.
| Quantity | Formula (level ground) |
|---|---|
| Horizontal velocity | vₓ = v₀ cos θ |
| Vertical velocity | vᵧ = v₀ sin θ |
| Time of flight | T = 2v₀ sin θ ÷ g |
| Maximum height | H = v₀² sin²θ ÷ 2g |
| Range | R = v₀² sin 2θ ÷ g |
When the launch point is higher than the landing point, the time of flight comes from solving the vertical motion: T = (vᵧ + √(vᵧ² + 2gh₀)) ÷ g. The calculator uses this general form, so it works for both cases. Our projectile motion guide explains where each formula comes from.
Worked Example
Worked example: a football kick
Question: A ball is kicked from the ground at 20 m/s and 30° above the horizontal. How far does it travel?
- vₓ = 20 cos 30° ≈ 17.32 m/s and vᵧ = 20 sin 30° = 10 m/s.
- Time of flight: T = 2 × 10 ÷ 9.81 ≈ 2.04 s.
- Range: R = 17.32 × 2.04 ≈ 35.3 m. Maximum height: H = 10² ÷ (2 × 9.81) ≈ 5.10 m.
Answer: The ball lands about 35.3 m away after 2.04 s, reaching a height of 5.10 m.
Why 45° Gives the Longest Range
On level ground the range depends on sin 2θ, which is largest when 2θ = 90°, so θ = 45°. Angles that add up to 90°, such as 30° and 60°, give the same range, but the higher angle keeps the projectile in the air for longer.
Tip: If you launch from a height, the best angle is slightly less than 45°, because the projectile already has extra time to fall. Try it in the calculator with a launch height of 10 m.
Practise: try these projectile motion MCQs for exam practice with full solutions.
Use this projectile motion calculator to check homework answers, compare launch angles or plan a real experiment. Because the projectile motion calculator shows each step, you can see exactly where the time of flight, height and range come from.
Frequently Asked Questions
Does this calculator include air resistance?
No. It uses the standard textbook model with no air resistance. Real objects, especially light ones, travel shorter distances than the calculator predicts.
What angle gives the maximum range?
On level ground, 45° gives the maximum range. When launching from a height, the best angle is a little lower than 45°.
Can I use this calculator for a horizontal launch?
Yes. Enter an angle of 0° and the launch height. The calculator then gives the time to hit the ground and how far the object travels horizontally.