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Projectile motion simulator
Set the launch, watch the arc — the numbers update with every slider move.
SIMULATOR · MATH · PHYSICS
Projectile Motion Simulator
Drag the sliders and watch the arc fly: 20 m/s at 45° from the ground travels 40.77 m, peaks at 10.19 m, and lands after 2.88 s. The dot animates along the real path, and you can save up to three launches to compare them side by side. No air resistance, just gravity at 9.81 m/s² — the clean physics underneath every throw, kick, and shot.
The 60-second rule
Split the launch into sideways and up-down parts, then let gravity do the rest. Horizontal speed never changes (nothing pushes sideways), while vertical speed drops by 9.81 m/s every second. At the defaults — 20 m/s at 45° from the ground — the split is 14.14 m/s each way, the flight lasts 2.88 s, and the range is 40.77 m.
The worked example, line by line
Defaults: 20 m/s, 45°, 0 m. Split: vx = 20 × cos 45° = 14.14 m/s, vy = 20 × sin 45° = 14.14 m/s. Flight time = 2 × 14.14 ÷ 9.81 = 2.88 s. Range = 14.14 × 2.88 = 40.77 m. Max height = 14.14² ÷ (2 × 9.81) = 10.19 m. Impact speed = 20.00 m/s — same as launch, because it lands at the height it left.
Branch — launched from height: 20 m/s, 45°, 10 m up. The extra fall stretches the flight to 3.47 s and the range to 49.08 m; peak height is 20.19 m and it lands at 24.42 m/s — faster than it left, because gravity had 10 m of extra falling to work with.
Branch — straight up: 10 m/s, 90°, 0 m. No sideways motion at all: it climbs to 5.10 m, hangs for a blink, and comes back down after 2.04 s, landing at 10.00 m/s.
Branch — low and fast: 25 m/s, 30°, 5 m up. Range 62.79 m, flight 2.90 s, peak 12.96 m, impact 26.89 m/s — the shallow angle trades height for distance.
Output · formula · meaning
| Output | Formula | What it means |
|---|---|---|
| Range | vx × T | 40.77 m at the defaults — how far from the launch point it lands. |
| Max height | h + vy² ÷ (2g) | 10.19 m at the defaults — the top of the arc, where vertical speed hits zero. |
| Flight time | (vy + √(vy² + 2gh)) ÷ g | 2.88 s at the defaults — up and back down to the ground. |
| Impact speed | √(vx² + vy² + 2gh) | 20.00 m/s at the defaults — faster than launch whenever it falls from height. |
| Horizontal velocity | v × cos θ | 14.14 m/s — constant from launch to landing. |
| Vertical velocity | v × sin θ | 14.14 m/s at launch, shrinking by 9.81 m/s each second. |
Who uses this
- The physics student: 20 m/s at 45° is the textbook case — 40.77 m, 2.88 s, 10.19 m — and the slider shows instantly why 45° wins on flat ground.
- The coach: shot put, javelin, long jump — the same arc decides every throw, and the height branch shows why a release from shoulder height beats one from the ground.
- The game developer: tuning a jump arc or a catapult — the simulator is the exact model most 2D physics uses before drag gets added.
- The hobby rocketeer: a model rocket's coast phase is a projectile problem; the flight-time and impact-speed numbers size the parachute delay.
Accuracy & limitations
This is the ideal vacuum model: constant gravity of 9.81 m/s², flat ground, no wind, no spin, no air resistance. Real throws land shorter — a baseball at 40 m/s loses a noticeable share of its range to drag, and a golf ball's dimples change the drag entirely. Treat the answers as the ceiling for a real throw, exact for a textbook problem. At 100 m/s the drag error dominates, so use the top of the slider for intuition, not for engineering.
Cite this calculatorBright Side Kit. “Projectile Motion Simulator.” brightoolbox.com, 2026.
Bright Side Kit. "Projectile Motion Simulator." https://brightoolbox.com/projectile-motion-simulator/ Accessed October 26, 2026.Frequently asked questions
What is the range of a projectile launched at 45 degrees?
From flat ground, range = v² ÷ g. At 20 m/s that is 400 ÷ 9.81 = 40.77 m. 45° is the farthest angle from level ground because it splits the speed evenly between forward motion and hang time.
How do you calculate flight time with launch height?
Flight time = (vy + √(vy² + 2gh)) ÷ g, where vy = v × sin θ and h is the launch height. At 20 m/s, 45°, 10 m up, that gives 3.47 s — the extra 10 m of falling adds about 0.59 s over the flat-ground 2.88 s.
What launch angle gives the maximum range?
45° from flat ground. From a height, the best angle is lower — at 10 m up with 20 m/s, the optimum is about 39° (49.78 m, versus 49.08 m at 45°). The simulator's save-and-compare shows this directly: save 45°, then try 39° and 40° from the same height.
Why does launch height change the range?
Because the projectile has farther to fall, it stays airborne longer, and the constant horizontal speed keeps carrying it forward the whole time. From 10 m up at 20 m/s and 45°, the range grows from 40.77 m to 49.08 m — about 20% farther for 10 m of height.
What is impact velocity?
The speed at the instant it hits the ground: √(vx² + vy² + 2gh). From flat ground it equals the launch speed (20.00 m/s at the defaults); from 10 m up it is 24.42 m/s, because gravity added speed during the extra fall.
Why does the simulator ignore air resistance?
Because the clean model is the one every textbook and most games use first — and the drag math needs the object's size, shape, and mass, which a simple simulator cannot know. The answers are exact for the ideal model and a ceiling for real throws: a real ball always lands shorter.