Projectile RangeFree online science calculator — instant results, no sign-up
Calculate the horizontal range of a projectile launched from ground level.
About the Projectile Range
The Projectile Range calculator estimates how far a projectile travels horizontally when launched from ground level and landing at the same height. Enter a launch velocity and angle to calculate its range, maximum height, and flight time under ideal conditions without air resistance.
- Launch velocity (m/s)
- The projectile’s initial speed at launch, measured in meters per second, with a default of 20 m/s.
- Launch angle (°)
- The projectile’s launch direction measured in degrees above the horizontal, with a default of 45°.
Formula and method
θ = α × π / 180; R = v₀² sin(2θ) / g; H = v₀² sin²(θ) / (2g); T = 2v₀ sin(θ) / g, using g = 9.80665 m/s².
- v₀Initial launch speed in meters per second.
- αLaunch angle above the horizontal in degrees.
- θLaunch angle converted to radians for the trigonometric calculations.
- πThe mathematical constant pi, used to convert degrees to radians.
- gConstant downward gravitational acceleration, taken as 9.80665 m/s².
- RHorizontal range from launch to landing, in meters.
- HMaximum vertical height above the launch level, in meters.
- TTotal time from launch to landing, in seconds.
Convert the launch angle from degrees to radians, then resolve the launch velocity into horizontal and vertical components: v₀ cos(θ) and v₀ sin(θ). Find the flight time from the vertical motion returning to ground level, multiply that time by the constant horizontal velocity to obtain range, and calculate maximum height from the initial vertical velocity. With 20 m/s and 45°, the results rounded to three decimal places are Range = 40.789 m; Max height = 10.197 m; Flight time = 2.884 s.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Launch velocity20 m/s
- Launch angle45 °
Results
- Range40.789 m
- Max height10.197 m
- Flight time2.884 s
How to use the Projectile Range
- 1Enter the launch velocity in m/s; keep the default of 20 to follow the worked example.
- 2Enter the launch angle in degrees above horizontal; keep the default of 45 for the worked example.
- 3Calculate the results using the entered speed and angle.
- 4Read the range, maximum height, and flight time; the default inputs give 40.789 m, 10.197 m, and 2.884 s, respectively.
- 5Adjust the speed or angle to compare ideal trajectories while keeping launch and landing at the same ground level.
Reading your result
Range is the horizontal distance between the launch and landing points, not the distance traveled along the curved trajectory; the default result is 40.789 m. Max height is the highest point above the launch level, which is 10.197 m for the default inputs. Flight time is the full time in the air, including ascent and descent, which is 2.884 s for the default inputs.
Limitations
The model assumes equal launch and landing elevations, flat ground, constant gravity of 9.80665 m/s², and no air resistance, wind, lift, or effects of spin. It treats the projectile as a point object and does not account for obstacles. For an ordinary above-ground flight, use a positive launch speed and an angle greater than 0° and less than 90°; a vertical 90° launch has zero horizontal range. Real trajectories, especially over long distances or at high speeds, can differ substantially. Displayed results are rounded.
Common questions
Why is 45° the default launch angle?+
For a fixed launch speed with equal launch and landing heights and no air resistance, range is proportional to sin(2θ). This factor reaches its maximum at 45°, so that angle produces the greatest horizontal range in this model.
Why do two different launch angles sometimes give the same range?+
Complementary angles—angles that add to 90°—have the same value of sin(2θ), so they give the same range at the same launch speed. The higher angle produces a greater maximum height and longer flight time.
Can I use this calculator for a projectile launched from a cliff?+
Not directly. The range and flight-time formulas assume the projectile lands at its launch elevation. A cliff launch requires the height difference to be included in the vertical-motion equation before calculating horizontal range.
Why might using 9.81 m/s² give slightly different default results?+
The worked example uses g = 9.80665 m/s². Replacing it with the rounded value 9.81 m/s² slightly changes all three outputs, so use the stated gravity value and round only the final results to reproduce Range = 40.789 m, Max height = 10.197 m, and Flight time = 2.884 s.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.