Projectile Motion Calculator
Calculate range, maximum height, time of flight, and velocity components for a projectile launched at any angle and speed. Assumes no air resistance.
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How to use this calculator
R = horizontal range; H = maximum height; T = total time of flight; v₀ = initial speed; θ = launch angle; g = 9.81 m/s².
- 1
Enter the initial speed and launch angle from horizontal.
- 2
Enter the launch height if the projectile starts above ground (default 0).
- 3
Gravity defaults to 9.81 m/s² (Earth). Change it to 1.62 for the Moon, 3.72 for Mars.
Frequently asked questions
Which angle gives the longest range?
On level ground with no air resistance it is 45°. If you launch from above the landing level, as with a ball thrown from a cliff, the best angle drops below 45°: tan θ = v₀ ÷ √(v₀² + 2gh₀). With a 20 m/s throw from a 10 m height, 40° reaches 49.770 m while 45° reaches 49.082 m, and the calculator confirms it.
What does the launch height field do?
It sets the starting height above the surface where the projectile lands (y = 0). The flight ends when the height returns to zero, so a launch from 50 m at 0° with 15 m/s flies 3.193 s and lands 47.891 m away. Maximum height is measured from that same zero level, so it includes the launch height.
Why do I get "Projectile never lands"?
This message appears when the launch height is negative and the projectile cannot climb back up to the zero level, for example 20 m/s at 45° from -100 m. Negative heights are only meaningful when the projectile can actually reach y = 0. Check the sign of the height field.
Can I use a negative or very steep angle?
Yes. A negative angle throws the object downward: 15 m/s at -30° from 50 m gives a vertical speed of -7.500 m/s and a flight time of 2.518 s, and the maximum height shown is just the 50 m starting height. At 90° the horizontal range is 0 and the object goes straight up and back down.
Does the landing speed depend on the angle?
Only through the launch height. When launch and landing levels are the same, the final speed equals the launch speed whatever the angle (20.000 m/s at 30°, 45° and 60° alike). From 10 m up, 35°, 40° and 45° all land at 24.417 m/s, because energy conservation fixes the speed at a given height.
How do I change gravity for another planet?
Edit the gravity field. At 1.62 m/s² (the Moon value suggested in the instructions) a 20 m/s, 45° launch travels 246.914 m for 17.459 s; at 3.72 m/s² (Mars) it travels 107.527 m for 7.603 s. The value must be above zero or the tool returns an error.
Does it include air drag, spin or wind?
No. The model is a point mass in uniform gravity and a vacuum. Drag shortens real ranges, and for fast or light objects the difference is large, so treat the output as an upper bound for anything flying through air.
Projectile motion calculator — range, peak height and flight time without air drag
Two independent motions
Horizontal velocity stays constant at v₀ cos θ while the vertical velocity changes by g every second. Flight time comes from solving h₀ + v_y t − ½gt² = 0 for the later root, and range is horizontal speed times that time. Launch height therefore matters for flight time, range and peak, but not for the launch speed components. The tool reports those components separately, so you can see how much of a throw is spent going forward and how much going up. At 45° the two are equal, 14.142 m/s each for a 20 m/s launch, and at 30° the horizontal share is 17.321 m/s against a vertical 10 m/s.
Worked example: 20 m/s at different angles
With g = 9.81 m/s² and a launch from the ground, the calculator returns the following. Range peaks at 45°, while the peak height and time in the air keep growing as the angle steepens.
| Angle | Range | Max height | Flight time |
|---|---|---|---|
| 15° | 20.387 m | 1.366 m | 1.055 s |
| 30° | 35.312 m | 5.097 m | 2.039 s |
| 45° | 40.775 m | 10.194 m | 2.883 s |
| 60° | 35.312 m | 15.291 m | 3.531 s |
| 75° | 20.387 m | 19.022 m | 3.939 s |
Complementary angles
The pairs 15° and 75°, and 30° and 60°, land at identical distances (20.387 m and 35.312 m above). The steeper shot spends more time in the air and climbs much higher, which is why a lob and a flat drive can reach the same spot. This symmetry only holds when launch and landing heights match. Raise the launch point and the pairing breaks, because the best angle for range moves below 45°.
Worked example: throwing off a ledge
Throw a ball horizontally at 15 m/s from 50 m. The calculator reports a flight time of 3.193 s, a range of 47.891 m and a landing speed of 34.728 m/s, of which 15 m/s is the unchanged horizontal part. Aiming 30° downward at the same speed shortens the flight to 2.518 s and the range to 32.716 m, yet the landing speed is still 34.728 m/s.
Where the model stops being accurate
Real projectiles lose speed to air, so range from the formulas is a ceiling. The model also assumes constant gravity over the whole flight and ignores the curvature and rotation of the Earth, which only matters over very long distances. Wind, spin and the lift on a rotating ball are also absent. For a quick sanity check the calculator is excellent, and for a competitive shot or a ballistic design it is the first approximation to refine rather than the final word.
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →