Projectile Motion Calculator
Enter how fast and at what angle something is launched, and from how high, to get how far it goes, how high it gets, how long it flies and how fast it lands, with a plot of its path. Advanced finds the angle or speed to hit a target, the position at any moment, landing on higher or lower ground, and Moon or Mars gravity. No air resistance.
Angle 0° is flat, 90° is straight up, below 0° is aimed downward. Mix units freely, for example mph with feet. No air resistance.
Hit a target
How far away is the target? Uses the speed, angle, heights and gravity above. The target sits at the landing height.
Where is it at a moment?
Seconds after launch. The point shows as a white dot on the plot.
Range at every angle
Your speed, heights and gravity at angles from 5° to 85°.
| Angle | Range | Top height | Flight time |
|---|
How your projectile is worked out
Your numbers: split the speed, find the time in the air, then the distance and the top of the arc.
- 1
Split the speed
20 m/s at 45°across 14.14 m/sup 14.14 m/s
- 2
Time in the air
2 × 14.14÷ 9.807= 2.884 s
- 3
Distance across
14.14 m/s× 2.884 s= 40.79 m
- 4
Highest point
14.14² ÷ 19.61= 10.2 m
Read more: the formulas, worked examples and the best angle
Once something is launched, only gravity acts on it (we leave out the air). Gravity pulls straight down, so it does nothing to the sideways speed: that stays the same all the way. The upward speed drops by g every second until it is zero at the top, then the projectile falls back faster and faster. Put the two together and you get the familiar curved path, a parabola.
The formula
speed across vx = v × cos θ
speed up vy = v × sin θ
height at time t: y = h + vy × t − ½ × g × t²
distance at time t: x = vx × t
time of flight t = (vy + √(vy² + 2 × g × (h − landing height))) ÷ g
range R = vx × t
time to the top = vy ÷ g
maximum height = h + vy² ÷ (2 × g)
impact speed = √(v² + 2 × g × (h − landing height))
on flat ground: R = v² × sin(2θ) ÷ g, t = 2 × vy ÷ g
angles to hit a target R: tan θ = (R ± √(R² − 4a(a + Δy))) ÷ (2a), a = g × R² ÷ (2v²), Δy = target height − h
speed for a range at angle θ: v = √(g × R² ÷ (2 cos² θ × (R tan θ − Δy)))
v in m/s, θ in degrees, h and R in m, g = 9.80665 m/s² on Earth
Worked example
The default: 20 m/s at 45° from the ground, landing on the same level, on Earth. Speed across: 20 × cos 45° = 14.1421 m/s; speed up: 20 × sin 45° = 14.1421 m/s. Time of flight: 2 × 14.1421 ÷ 9.80665 = 2.8842 s. Range: 14.1421 × 2.8842 = 40.789 m (check: 20² × sin 90° ÷ 9.80665 = 400 ÷ 9.80665 = 40.789 m). Time to the top: 14.1421 ÷ 9.80665 = 1.4421 s. Maximum height: 14.1421² ÷ (2 × 9.80665) = 200 ÷ 19.6133 = 10.197 m. It lands at 20 m/s, 45° below flat, the mirror image of the launch.
From a height: 20 m/s at 30° from 10 m up. Speed across 17.3205 m/s, up 10 m/s. Time of flight: (10 + √(10² + 2 × 9.80665 × 10)) ÷ 9.80665 = (10 + √296.133) ÷ 9.80665 = (10 + 17.2085) ÷ 9.80665 = 2.7745 s. Range: 17.3205 × 2.7745 = 48.056 m. Maximum height: 10 + 10² ÷ 19.6133 = 15.099 m. Impact speed: √(400 + 196.133) = 24.416 m/s, at 44.81° below flat.
Position at 1 s (the default): across 14.1421 × 1 = 14.142 m, height 14.1421 × 1 − ½ × 9.80665 × 1² = 14.1421 − 4.9033 = 9.239 m, still rising at 14.1421 − 9.80665 = 4.335 m/s.
Hit a target 30 m away at 20 m/s (flat ground): sin 2θ = 9.80665 × 30 ÷ 400 = 0.73550, so 2θ = 47.349° or 132.651°, giving 23.67° (low) or 66.33° (high). They add up to 90°. The furthest reach at 20 m/s is 40.789 m at 45°.
Speed for 50 m at 45° (flat ground): v = √(9.80665 × 50 ÷ sin 90°) = √490.3325 = 22.143 m/s (79.72 km/h).
On the Moon (g ≈ 1.62 m/s²) the default throw goes 400 ÷ 1.62 = 246.9 m and climbs 200 ÷ 3.24 = 61.7 m, about 6 times further and higher than on Earth.
Range at every angle (20 m/s, flat ground, Earth)
| Angle | Range | Top height | Flight time |
|---|---|---|---|
| 5° | 7.083 m | 0.1549 m | 0.3555 s |
| 10° | 13.95 m | 0.615 m | 0.7083 s |
| 15° | 20.39 m | 1.366 m | 1.056 s |
| 20° | 26.22 m | 2.386 m | 1.395 s |
| 25° | 31.25 m | 3.643 m | 1.724 s |
| 30° | 35.32 m | 5.099 m | 2.039 s |
| 35° | 38.33 m | 6.71 m | 2.34 s |
| 40° | 40.17 m | 8.426 m | 2.622 s |
| 45° (best) | 40.79 m | 10.2 m | 2.884 s |
| 50° | 40.17 m | 11.97 m | 3.125 s |
| 55° | 38.33 m | 13.68 m | 3.341 s |
| 60° | 35.32 m | 15.3 m | 3.532 s |
| 65° | 31.25 m | 16.75 m | 3.697 s |
| 70° | 26.22 m | 18.01 m | 3.833 s |
| 75° | 20.39 m | 19.03 m | 3.94 s |
| 80° | 13.95 m | 19.78 m | 4.017 s |
| 85° | 7.083 m | 20.24 m | 4.063 s |
Angles the same distance either side of 45° (30° and 60°, 15° and 75°) reach the same distance, but the higher one climbs higher and stays up longer.
Gravity on other worlds
| Where | g | Range of 20 m/s at 45° |
|---|---|---|
| Earth | 9.80665 m/s² | 40.79 m |
| Moon (approximate) | ≈ 1.62 m/s² | 246.9 m |
| Mars (approximate) | ≈ 3.71 m/s² | 107.8 m |
Notes
- No air resistance. Real objects slow down in the air, so they go shorter and lower than shown, and their best angle is usually under 45°. The difference is small for a heavy ball thrown gently and large for a golf ball, an arrow or a bullet.
- Heights are measured from the launch ground. The launch height is where it leaves the hand or barrel; a landing height above 0 means higher ground (a roof or a hill), below 0 means lower ground (a valley or off a cliff).
- The Earth is treated as flat and gravity as the same everywhere, which is fine for anything that stays within a few kilometres.
- Standard gravity g = 9.80665 m/s² is a defined value; real gravity is about 9.78 m/s² at the equator and 9.83 m/s² at the poles. Moon (≈ 1.62 m/s²) and Mars (≈ 3.71 m/s²) are rounded surface averages.
- Unit factors are exact: 1 ft = 0.3048 m, 1 yd = 0.9144 m, 1 mph = 0.44704 m/s, 1 km/h = 1 ÷ 3.6 m/s, 1 knot = 1,852 m per hour.
Common questions
What is projectile motion?
Projectile motion is the curved path of anything thrown, kicked or fired that then flies with only gravity acting on it. Its sideways speed stays the same, while gravity steadily slows its climb and then speeds up its fall, so the path is a parabola. This calculator leaves out air resistance.
What is the formula for the range of a projectile?
On flat ground (launched and landing at the same height) the range is R = v² × sin(2θ) ÷ g. At 20 m/s and 45° on Earth that is 400 × 1 ÷ 9.80665 = 40.79 m. From a height h, the time of flight is t = (v sin θ + √((v sin θ)² + 2gh)) ÷ g and the range is v cos θ × t.
How do I find the maximum height?
Maximum height = h + (v sin θ)² ÷ (2g), where h is the launch height. At 20 m/s and 45° from the ground: (14.142)² ÷ 19.6133 = 10.197 m. The top is reached after v sin θ ÷ g seconds, 1.442 s here.
Why is 45° the best angle?
On flat ground the range depends on sin(2θ), which is biggest (1) when 2θ = 90°, so θ = 45°. Angles the same distance either side of 45°, like 30° and 60°, give the same range. From a height the best angle is lower than 45°: from 10 m at 20 m/s it is about 39.3°.
Why are there two angles to hit the same target?
For any target closer than the furthest reach there is a flat, fast shot and a high, lobbed shot that both land there. On flat ground the two angles add up to 90°: at 20 m/s, a target 30 m away can be hit at 23.67° or 66.33°. The high shot takes longer and goes higher.
Does this include air resistance?
No. Every answer assumes no air resistance (a vacuum), which is close for heavy, slow, compact objects over short distances. Real balls, arrows and bullets go shorter and lower, land steeper, and their best angle is usually below 45°.
What gravity does it use?
Standard Earth gravity g = 9.80665 m/s² by default. In Advanced you can pick the Moon (about 1.62 m/s²), Mars (about 3.71 m/s²) or type your own. Less gravity means a longer, higher, slower flight: on the Moon the same throw goes about 6 times as far.
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