Projectile Motion Calculator (range, height, flight time)
Work out the range, peak height, flight time, and landing speed of a projectile from its launch speed, angle, and height.
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Range by launch angle
Launching at 20 m/s from ground level, ignoring air resistance. Notice that angles either side of 45° pair up — 30° and 60° travel exactly the same distance, one on a flat fast arc and one on a high slow one.
| Launch angle | Range | Peak height | Flight time |
|---|---|---|---|
| 10° | 13.95 m | 0.615 m | 0.708 s |
| 15° | 20.39 m | 1.366 m | 1.06 s |
| 20° | 26.22 m | 2.386 m | 1.4 s |
| 25° | 31.25 m | 3.643 m | 1.72 s |
| 30° | 35.32 m | 5.099 m | 2.04 s |
| 35° | 38.33 m | 6.71 m | 2.34 s |
| 40° | 40.17 m | 8.426 m | 2.62 s |
| 45° | 40.79 m | 10.2 m | 2.88 s |
| 50° | 40.17 m | 11.97 m | 3.12 s |
| 55° | 38.33 m | 13.68 m | 3.34 s |
| 60° | 35.32 m | 15.3 m | 3.53 s |
| 70° | 26.22 m | 18.01 m | 3.83 s |
| 80° | 13.95 m | 19.78 m | 4.02 s |
Projectile motion
Once something is in the air with nothing pushing it, its motion splits cleanly into two independent problems. Horizontally it just keeps going at a constant speed. Vertically it is infree fall, slowing on the way up and speeding up on the way down. Solving the two separately and combining them is the whole technique.
x = v₀·cosθ·t · y = h₀ + v₀·sinθ·t − ½g·t²
From those two lines everything else follows: the flight time is when y returns to zero, the range is the horizontal distance covered in that time, and the peak is where the vertical velocity passes through zero. This calculator reports all of them, plus the speed and angle at which the projectile lands.
Worked example
Thrown at 20 m/s at 45° from ground level, a ball travels 20² × sin(90°) / 9.807 ≈ 40.8 m, peaks at about 10.2 m, and is in the air for roughly 2.9 s. Throw it at 30° instead and the range drops to about 35.3 m — and so does throwing it at 60°, because complementary angles always give the same distance.
Why 45° is the best angle — usually
Range depends on the product of horizontal speed and time in the air. A shallow angle gives plenty of horizontal speed but very little hang time; a steep one gives the opposite. From ground level the compromise lands exactly at 45°. Launch from a height, though, and the balance shifts: you are already getting free hang time from the drop, so a slightly flatter angle wins. Set a launch height above zero and the calculator works out the best angle for you.
Landing speed doesn't depend on the angle
A projectile launched from a height lands at the same speed whatever angle you choose — only the direction changes. Energy explains it: the speed on landing comes from the launch speed plus whatever the drop adds, and neither cares about the angle. Try 10° and 70° from a 12 m launch height and the landing speeds match exactly.
The air-resistance caveat
These equations ignore drag, which real projectiles very much do not. For a thrown ball over short distances the error is modest; for a bullet, a badminton shuttle, or anything light and fast it is enormous, and the real trajectory becomes markedly asymmetric — falling more steeply than it rose. See terminal velocity for what drag does to a falling object.