Projectile Motion Calculator
Calculate projectile motion: range, max height, time of flight and a trajectory chart. Enter launch velocity, angle, initial height and gravity.
How to use this tool
- Enter launch velocity, launch angle, initial height and gravitational acceleration in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your horizontal range and the full breakdown beneath it.
Projectile motion splits velocity into horizontal (v₀cosθ) and vertical (v₀sinθ) components. Horizontal velocity is constant; vertical is affected by gravity g = 9.81 m/s².
- Range: R = v₀²sin(2θ)/g (from ground level)
- Max height: H = v₀²sin²θ / (2g)
- Time of flight: T = 2v₀sinθ / g
Formula
Horizontal: x(t) = vx × t | vx = v0 cos(θ)
Vertical: y(t) = h0 + vyt − ½gt2 | vy = v0 sin(θ)
Time of flight: tf = (vy + √(vy2 + 2gh0)) / g
Range = vx × tf | Max height = h0 + vy2 / (2g)
How it works
The calculator decomposes the launch velocity into horizontal (v cos θ) and vertical (v sin θ) components, then applies Newton's equations of motion under constant gravity. Time of flight is found by solving the quadratic y(t) = 0. The model assumes no air resistance, a flat surface, and constant gravitational acceleration — all reasonable for short-range, low-speed trajectories but increasingly inaccurate for high-speed or long-range projectiles where drag is significant.
Worked example
- Launch velocity v₀ = 20 m/s, angle θ = 45°, initial height h₀ = 0 m, gravity g = 9.81 m/s².
- vx = 20 × cos(45°) = 20 × 0.7071 = 14.142 m/s; vy = 20 × sin(45°) = 14.142 m/s.
- Time of flight: tf = (14.142 + √(14.142² + 0)) / 9.81 = (14.142 + 14.142) / 9.81 = 28.284 / 9.81 ≈ 2.884 s.
- Horizontal range = 14.142 × 2.884 ≈ 40.77 m.
- Max height = 0 + 14.142² / (2 × 9.81) = 200 / 19.62 ≈ 10.19 m.
Horizontal range: 40.77 m; Maximum height: 10.19 m; Time of flight: 2.884 s.
Common mistakes to avoid
- Entering launch angle in radians when the calculator expects degrees — 45 radians is an invalid angle for typical problems, but the formula will still compute a nonsensical trajectory.
- Mixing unit systems — using g=9.8 m/s^2 with velocity in feet per second produces wrong range and height values; all inputs must use consistent units.
- Expecting maximum range at exactly 45 degrees when launching from an elevated initial height — the optimal angle shifts below 45 degrees when h0 > 0.
Key terms
- Launch angle (θ)
- The angle above the horizontal at which the projectile is fired. 45° maximises range on level ground when launched from ground level.
- Horizontal range
- The total horizontal distance travelled from launch to landing, equal to the horizontal speed multiplied by the time of flight.
- Time of flight
- The total time the projectile spends in the air, from launch until it returns to the ground (or the initial height level if h₀ = 0).
- Maximum height
- The peak vertical position reached by the projectile, occurring at the moment the vertical velocity component equals zero.
- Air resistance (drag)
- A retarding force proportional to speed that is not modelled here. In reality, drag reduces range and maximum height, especially for fast or light projectiles.
Frequently asked questions
- Which angle maximises range?
- 45° maximises range on flat ground. With an elevated launch point the optimal angle is slightly less than 45°.
- Can I use different gravity values?
- Yes — use 1.62 m/s² for the Moon, 3.72 for Mars, or 24.79 for Jupiter.