Projectile Motion Simulator
A cannon, four gravities, and the real equations running live. Dial in angle and speed, watch the exact parabola fly, flip on air resistance to see the textbook break — and hit targets for score.
What is the Projectile Motion Simulator?
This is a physics lab disguised as a cannon game. You choose a launch angle, a launch speed, and one of four gravities — the Moon, Mars, Earth or Jupiter — and the simulator flies the shot with the real equations of motion. The three numbers every physics course cares about (range, maximum height, flight time) are computed from the exact closed-form formulas and shown with your values plugged in, so you can watch the algebra and the animation agree in real time.
The air-resistance toggle is where it gets honest. The tidy parabola in every textbook assumes a vacuum; flip the toggle and the simulator integrates the flight step by step with quadratic drag instead, drawing both curves so you can see exactly how far reality falls short of the ideal — and why the best real-world throwing angle is flatter than the famous 45 degrees. There is also a target round: ten shots, points for accuracy, best score saved.
How to Use It
The Physics, Written Out
Three formulas run this whole page. For launch speed v, angle θ and gravity g (ignoring air):
Range: R = v²·sin(2θ) ÷ g. The sin(2θ) term is why 45° throws furthest in a vacuum — sin(90°) = 1 is its peak — and why 30° and 60° land in exactly the same place.
Maximum height: H = v²·sin²θ ÷ 2g. Height only cares about the vertical part of the launch, which is why a steep 80° shot climbs enormously but lands almost at your feet.
Flight time: T = 2v·sinθ ÷ g. Double the speed and you double the time aloft; go to the Moon and its g of 1.62 m/s² keeps the same shot airborne six times longer than Earth's 9.81.
A worked example: v = 50 m/s at 45° on Earth gives R = 2500 × 1 ÷ 9.81 ≈ 255 m, H = 2500 × 0.5 ÷ 19.62 ≈ 64 m, and T = 100 × 0.707 ÷ 9.81 ≈ 7.2 s. Set those sliders and the simulator's cards show exactly these numbers.
The Four Gravities
| World | g (m/s²) | Same shot vs Earth |
|---|---|---|
| 🌙 Moon | 1.62 | ≈ 6.1× the range, 6.1× the flight time |
| 🔴 Mars | 3.71 | ≈ 2.6× the range |
| 🌍 Earth | 9.81 | the baseline |
| 🟠 Jupiter | 24.79 | ≈ 0.4× — shots slam down fast |
Things to Try
Four experiments, each with a real physics lesson inside:
Prove 45° is king — in a vacuum
Fix the speed, fire at 30°, 45° and 60°, and compare ranges. The 45° shot wins, and the 30° and 60° shots land on the same spot — sin(60°) equals sin(120°). This complementary-angle pairing is a classic exam question you can now simply watch.
Then dethrone it with drag
Turn air resistance on and repeat. The best angle slides down to roughly 35–40°, because drag taxes every second spent in the air and high shots spend the most. This is why javelins, long throws and artillery all fly flatter than the textbook optimum.
Take the same shot to the Moon
Fire a 50 m/s shot on Earth, note the range, then switch to the Moon and fire again — about six times further, in exact proportion to the gravity ratio 9.81 ÷ 1.62. One slider press turns a constant from a table into something you can feel.
Chase the target efficiently
In the target round, resist re-aiming from scratch each shot. Range scales with v², so a small speed change moves the landing point a lot; fine-tune with speed and keep the angle near 40°. That sensitivity — quadratic in v, gentle in θ near the peak — is the actual shape of the range formula.
Honest Limits
The vacuum flight is exact — the same closed-form solution your physics textbook derives. The air-resistance flight uses quadratic drag with a fixed coefficient (a 0.005 per-metre drag-to-mass ratio, integrated at 240 steps per second), which behaves roughly like a tennis ball — a beach ball would sag far more, a cannonball far less. There is no wind, no spin and no Magnus effect, launches happen from ground level, and the ground is flat. Every one of those simplifications keeps the two curves comparable — one ideal, one dragged, nothing else changing.
FAQ
Is the Projectile Motion Simulator free?
Yes — the lab is completely free and runs instantly in your browser. There is no download, no sign-up, and nothing to install.
Are the physics real?
Yes. The ideal trajectory is the exact closed-form solution — range v²·sin(2θ)/g, height v²·sin²θ/(2g), time 2v·sinθ/g — with your numbers plugged in live. The air-resistance flight is integrated step by step with quadratic drag, which is why it always lands short of the ideal curve, exactly as a real ball does.
Why is 45 degrees the best launch angle?
In a vacuum, range is proportional to sin(2θ), which peaks when 2θ = 90° — that is, at 45°. Turn air resistance on and the answer changes: drag punishes time spent in the air, so the best angle drops to roughly 35–40°, which is why real long throws are flatter than the textbook answer.
What do the different gravities change?
Everything scales with g. The simulator offers the Moon (1.62 m/s²), Mars (3.71 m/s²), Earth (9.81 m/s²) and Jupiter (24.79 m/s²). The same shot flies about six times further on the Moon than on Earth, because range divides by g — a shot that lands at 100 m here would carry about 600 m there.
Is my target score saved?
Yes. Your best score from the 10-shot target round is stored in your browser's local storage on this device. No account is needed, and clearing your browser data resets it.