Projectile Motion Calculator
A projectile follows a parabola set entirely by its launch speed and angle. This calculator returns range, peak height and time in the air.
Formula
Range = v² sin(2θ) ÷ g · Max height = (v sin θ)² ÷ 2g · Flight = 2v sin θ ÷ g
- v = launch speed, θ = launch angle, g = 9.81 m/s²
- Ignores air resistance, which matters a great deal in practice
20 m/s at 45°: range 40.8 m, peak 10.2 m, 2.88 s in the air
The Formulas
- Range: R = v² sin(2θ) ÷ g
- Maximum height: H = v² sin²θ ÷ 2g
- Time of flight: T = 2v sinθ ÷ g
Launched at 20 m/s and 45°: range 40.8 m, peak height 10.2 m, airborne for 2.88 s.
Why 45 Degrees Is Optimal
Range depends on sin(2θ), which peaks when 2θ = 90° — that is, at 45°.
A useful consequence: angles either side of 45° that sum to 90° give the same range. A 30° launch and a 60° launch travel equally far, the first flat and fast, the second high and slow.
Horizontal and Vertical Are Independent
The key insight is that the two directions do not interact. Horizontal velocity stays constant; vertical velocity changes under gravity alone.
This is why a bullet fired horizontally and one simply dropped from the same height hit the ground at the same moment, however far the fired one travels.
These formulas ignore air resistance, which is fine for a shot put and badly wrong for a football or an arrow. Real sporting ranges fall well short of the ideal figures, and optimum angles drop below 45°.
Quick Reference
| Angle | Range | Max height | Time of flight |
|---|---|---|---|
| 15° | 20.4 m | 1.4 m | 1.06 s |
| 30° | 35.3 m | 5.1 m | 2.04 s |
| 45° | 40.8 m | 10.2 m | 2.88 s |
| 60° | 35.3 m | 15.3 m | 3.53 s |
| 75° | 20.4 m | 19.0 m | 3.94 s |
Frequently Asked Questions
- What angle gives the greatest range?
- 45 degrees, in the absence of air resistance. Angles that sum to 90 degrees, such as 30 and 60, give identical ranges.
- Does a heavier projectile travel further?
- Not in a vacuum: mass cancels out of the equations. In air, heavier objects are less affected by drag, so they usually do travel further.
- Do horizontal and vertical motion affect each other?
- No. They are independent, which is why a bullet fired horizontally lands at the same time as one dropped from the same height.
- Why do real throws fall short of the calculation?
- These formulas ignore air resistance. Drag reduces range substantially for light or fast objects and lowers the optimum launch angle below 45 degrees.