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Science

Projectile Motion Calculator

Enter the initial speed v₀, launch angle θ (0° to 90°) and starting height h₀ — get instant values for horizontal range R, peak height H, total flight time T, landing speed and time-to-apex. Seven gravity presets (Earth, Moon, Mars, Venus, Jupiter, Mercury, Saturn) let you compare trajectories on other worlds. Internals are SI; pick speed in m/s, km/h, mph or ft/s and length in metres or feet.

Range R

Max height H

Flight time T

Formula (no air resistance)

R = vₓ · T · H = h₀ + vᵧ² ÷ 2g · T = (vᵧ + √(vᵧ² + 2gh₀)) ÷ g

Uses the idealised projectile-motion model: no drag, uniform gravity, point mass. Origin at the launch point with the y-axis pointing up.

Formula

Origin at launch point, y-axis upward: vₓ = v₀ · cos θ // horizontal velocity (constant) vᵧ = v₀ · sin θ // initial vertical velocity T = (vᵧ + √(vᵧ² + 2 g h₀)) ÷ g // flight time (solve y(T) = 0) R = vₓ · T // horizontal range H = h₀ + vᵧ² ÷ (2 g) // peak height above ground |v_land| = √(vₓ² + (vᵧ − g T)²) // speed at landing For h₀ = 0: R = v₀² · sin 2θ ÷ g, T = 2 v₀ sin θ ÷ g, and the range is maximised at θ = 45°. For h₀ > 0 the optimum angle is below 45° and satisfies tan θ* = v₀ / √(v₀² + 2 g h₀).

Frequently asked

Why is 45° the best angle for range from level ground? Is it still 45° when you launch from a rooftop?

From level ground (h₀ = 0), the range is R = v₀² sin 2θ ÷ g, and sin 2θ peaks at θ = 45°. When you launch from a rooftop or cliff (h₀ > 0), the extra fall gives "free" flight time and the optimum tilts toward more horizontal velocity, so the best angle drops below 45°. It satisfies tan θ* = v₀ / √(v₀² + 2 g h₀). For example, throwing 20 m/s from a 10 m roof on Earth, the optimum is about 41.6° — not 45°.

How accurate is the no-air-resistance answer for real life?

For dense, slow-moving objects (shot put, marbles, basketballs at < 20 m) the error is small, and this is the model textbooks and HKDSE use. For light, fast objects (badminton shuttles, ping-pong balls, baseballs) drag changes the picture a lot — a 40 m/s baseball travelling 100 m falls 30–40% short of the no-drag prediction. This tool is great for homework, exam prep and rough estimates, not for ballistics or competitive sport analysis.

How much farther can I throw on the Moon for the same launch?

The Moon's surface gravity is about g ≈ 1.62 m/s², roughly 1/6 of Earth's 9.81 m/s². For the same launch from ground level, range scales as 1/g, so the Moon range is about 9.81 ÷ 1.62 ≈ 6.06× the Earth value. Peak height and flight time scale by the same factor. Pick "Moon" in the gravity selector to see this directly — it's how people estimate Alan Shepard's famous Apollo 14 lunar golf shot.

Can I use this for HKDSE Physics projectile-motion problems?

Yes. The HKDSE Physics topic on projectile motion (Mechanics, Compulsory Part) uses exactly this no-air-resistance model. The tool gives you R, H, T and the landing speed in one shot. In the exam itself, draw the free-body diagram, write out vₓ and vᵧ, and substitute carefully — use this tool to cross-check the final numbers. Keep gravity on "Earth (9.81 m/s²)" by default; switch to "Moon" if the question is set on the lunar surface.

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