Projectile Range

🤔 How to play

Pick a level. Each one tells you what is fixed and what you choose: the angle, the speed, or both.

Set the launch angle θ and the launch speed v with the sliders, the − and + buttons or the keys.

Launch. The ball flies on its true path and drops a dot every tenth of a second or so, so you can see how it moves.

Read the result: how long it flew, how high it went, where it landed and how far it was from the target. Tap 📐 Working to see the formulas with your numbers in them.

Adjust and try again. Later levels add walls to clear, raised platforms, two-angle puzzles, the Moon, Mars and a heavy planet, and finally just three shots at narrow targets.

Keys: ← → change the angle by 1° and ↑ ↓ the speed by 1 m/s; hold Shift for 0.5° and 0.1 m/s steps. Enter or Space launches (and skips to the landing during a flight). H or 💡 gives a hint, R or 🎥 resets the view, W shows the working and P or 🔮 turns on predictions: type where you think the ball will land, and get a bonus tick if you are within 5%. Drag the scene to look around.

The physics

The ball starts at ground level with speed v at angle θ. With no air resistance the sideways speed v cos θ never changes, while gravity g takes g m/s off the upward speed v sin θ every second. On flat ground that gives:

Time of flight: T = 2v sin θ / g

Maximum height: H = v² sin² θ / 2g

Range: R = v² sin 2θ / g, longest at 45°, and the same for θ and 90° − θ

Height at any distance: y = x tan θ − g x² / (2v² cos² θ), for walls and platforms

Gravity is 9.8 m/s² on Earth, 1.6 on the Moon, 3.7 on Mars and 24.8 on the heavy planet. The steps are 0.5° and 0.1 m/s, and every level has been checked by computer to have settings on those steps that hit. Nothing is saved: ticks for finished levels last only this visit.

What it practises

Projectile motion as taught at GCSE and A level: splitting a velocity into horizontal and vertical parts, the equations of motion under constant gravity, the range equation and why 45° goes furthest, complementary angles, and how the path changes with gravity. Predicting first, then checking against a measured result, is the habit of a scientist.