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Projectile Range Calculator

Find how far a projectile travels.

How it is calculated

The model is an ideal projectile on level ground with no air resistance, launched and landing at the same height. R = v² · sin(2θ) / g gives the horizontal range, H = v² · sin²(θ) / (2g) the peak height above the launch point, and T = 2 · v · sin(θ) / g the total time in the air. Here v is the launch speed in metres per second, θ the launch angle in degrees above the horizontal, and g is fixed at 9.80665 m/s². Results are shown in metres and seconds with two decimals. The speed must be greater than zero, but the angle is not restricted, so angles outside 0 to 90 degrees are calculated mechanically and can give negative values that do not describe a real throw.

Worked example

A ball is launched at 20 m/s at 30 degrees. Range: sin(60°) = 0.8660, so R = 400 × 0.8660 / 9.80665 = 35.32 m. Peak height: sin²(30°) = 0.25, so H = 400 × 0.25 / (2 × 9.80665) = 5.10 m. Time of flight: T = 2 × 20 × 0.5 / 9.80665 = 2.04 s. Launching at 60 degrees with the same speed also gives a 35.32 m range, but the ball climbs to 15.30 m and stays up for 3.53 s.

Try this example →

Common mistakes

  • Entering the angle in radians. The field expects degrees, so 0.5 would be treated as half a degree, not about 29 degrees.
  • Entering speed in km/h or mph. The formulas assume metres per second, so divide km/h by 3.6 first.
  • Using the result for a launch from a cliff, a table or a raised platform. The model assumes the landing height equals the launch height.
  • Expecting air drag to be included. Real balls, arrows and water jets fly shorter than the ideal range, especially at high speed.
  • Typing an angle above 90 or a negative one. The tool does not reject it and can return a negative range or time.

About this calculator

For a projectile launched with speed v at angle θ over level ground (ignoring air resistance), range R = v²sin(2θ)/g, maximum height h = v²sin²θ/(2g), and time of flight t = 2v·sinθ/g, where g is gravitational acceleration (9.81 m/s² on Earth). Range is maximized at a 45° launch angle, and any two angles that sum to 90° (like 30° and 60°) produce the same range — a classic result from this formula.

Physics students use this to master projectile motion — decomposing velocity into horizontal and vertical components and treating them independently. It also has real applications: sports scientists analyze the launch angles of a thrown javelin or shot put, ballistics and artillery calculations historically relied on this exact model, and engineers use it for water-jet or fire-hose trajectory planning.

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Comments (1)

  • Editorial team

    Use the output as an upper bound, since air drag, spin and a different landing height all shorten or change the real flight. At 20 m/s the tool gives 35.32 m at both 30 and 60 degrees, but 60 degrees stays up for 3.53 s against 2.04 s. Complementary angles share a range, which is handy for checking your inputs. What are you launching, and from what height?

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