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Projectile Motion Calculator

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Enter launch speed and angle to find the range, maximum height, and time of flight of a projectile - ignoring air resistance.

Projectile Motion Calculator

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Range
40.82 m
Max height
10.2 m
Time of flight
2.89 s
Assumes launch and landing at the same height, and no air resistance - real projectiles (especially light or irregularly shaped ones) fall short of this ideal.

The formulas

Time of flight = (2 x Launch speed x sin(angle)) / g Max height = (Launch speed x sin(angle))^2 / (2 x g) Range = (Launch speed^2 x sin(2 x angle)) / g
Example

Launched at 20 m/s at a 45° angle: time of flight ≈ 2.89s, max height ≈ 10.2m, range ≈ 40.82 meters.

Step-by-step guide

  1. Enter the launch speed and the angle above horizontal.
  2. Read the range, maximum height, and time of flight.

Why 45 degrees gives the maximum range

For a given launch speed, 45° produces the longest possible range - this comes directly from the range formula, since sin(2 x angle) reaches its maximum value of 1 exactly when the angle is 45° (because 2 x 45° = 90°, and sine peaks at 90°). Interestingly, any two angles that add up to 90° (like 30° and 60°) produce the exact same range as each other, just with different heights and flight times - a classic, elegant symmetry in projectile motion.

Common mistakes

Entering the angle in radians instead of degrees - this calculator expects degrees, and using radians directly would give a wildly incorrect result.
Expecting this idealized result to exactly match a real thrown or launched object - air resistance meaningfully shortens real-world range and height, especially for lighter or less aerodynamic projectiles.

Frequently asked questions

Does this account for air resistance?

No - this uses the classic idealized projectile motion equations, which ignore air resistance entirely. Real-world results (especially for light objects or long distances) fall somewhat short of this ideal.

What if launch and landing heights are different?

This calculator assumes launch and landing occur at the same height, the standard simplified case. A projectile launched from an elevated point (like off a cliff) needs a more complete equation accounting for that extra height.

Why do 30° and 60° give the same range?

Because sin(2 x 30°) = sin(60°) and sin(2 x 60°) = sin(120°), and sin(60°) equals sin(120°) exactly - any pair of complementary angles (adding to 90°) produces identical range, a direct mathematical consequence of the range formula.

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