Projectile Motion Calculator — Range, Height, and Time
Calculate projectile motion: range, max height, flight time, and velocity components. Enter launch speed and angle to solve any projectile problem instantly.
Throw a ball, fire a cannon, kick a soccer ball — all of these are projectile motion problems. The key insight is that horizontal and vertical motion are completely independent. Gravity pulls the object down at 9.81 m/s², but it doesn't slow horizontal speed at all (ignoring air resistance). Treating these as two separate one-dimensional problems is what makes projectile motion tractable.
The CalcHub projectile motion calculator computes trajectory, max height, range, and flight time from any launch conditions.
Key Equations
Given launch velocity v₀ at angle θ above horizontal:
| Quantity | Formula |
|---|---|
| Horizontal velocity | vₓ = v₀ cos(θ) |
| Vertical velocity | vᵧ = v₀ sin(θ) |
| Max height | H = v₀² sin²(θ) / (2g) |
| Time of flight | T = 2v₀ sin(θ) / g |
| Range | R = v₀² sin(2θ) / g |
What Happens at Different Angles
| Launch Angle | Range | Max Height | Flight Time |
|---|---|---|---|
| 15° | Moderate | Low | Short |
| 30° | High | Medium | Medium |
| 45° | Maximum | Medium | Medium |
| 60° | Same as 30° | Higher | Longer |
| 75° | Same as 15° | Very high | Long |
How to Use the Calculator
- Enter initial velocity (m/s or km/h or mph)
- Enter launch angle (degrees above horizontal)
- Optionally set launch height if not at ground level
- Get range, max height, flight time, and velocity at any point
Worked Example
A football is kicked at 25 m/s at 40° above horizontal. How far does it travel?
R = 25² × sin(80°) / 9.81 = 625 × 0.985 / 9.81 ≈ 62.7 m
Max height: H = 25² × sin²(40°) / (2 × 9.81) = 625 × 0.413 / 19.62 ≈ 13.2 m
Flight time: T = 2 × 25 × sin(40°) / 9.81 = 50 × 0.643 / 9.81 ≈ 3.27 s
Does air resistance significantly change projectile motion?
For slow objects over short distances, not much. For fast projectiles (bullets, baseballs, golf balls) or anything over a long range, air resistance is significant — the actual range can be 20–50% shorter than the vacuum formula predicts. Ballistic calculators for firearms include drag coefficients for this reason.
Why does angle of 45° give maximum range?
The range formula R = v₀² sin(2θ) / g is maximized when sin(2θ) = 1, which happens when 2θ = 90°, so θ = 45°. It's a clean result that comes directly from the calculus of the range equation.
What if the projectile lands at a different height than it launched?
If landing height differs from launch height, the symmetric equations above don't apply. The calculator handles this case by solving the full quadratic for time of flight.