Select horizontal range, initial speed, launch angle, or gravity, then enter the known values. The ideal model assumes equal launch and landing heights and no air resistance.
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Catapult Range Formula
The calculator uses the standard projectile range equation for a launch and landing at the same height, without air resistance. Choose the value to solve. Gravity starts at a disclosed rounded Earth default of 9.81 m/sยฒ.
- R = horizontal range
- v = initial velocity
- θ = launch angle above horizontal
- g = acceleration due to gravity
- sin = sine function, with the angle converted to radians for the calculation
To calculate range, the calculator uses the entered velocity, launch angle, and gravity. To calculate initial velocity, it rearranges the range equation and solves for speed. To calculate launch angle, it shows both complementary inverse-sine solutions (one at 45 degrees for maximum range). To calculate gravity, it rearranges the range equation and solves for acceleration.
Launch Angle and Gravity Reference Values
The range depends strongly on the launch angle through the term sin(2θ). For the same initial velocity and gravity, a 45 degree launch gives the maximum range in this simplified model.
| Launch angle | sin(2θ) | Range compared with 45 degrees |
|---|---|---|
| 15° | 0.500 | 50.0% |
| 30° | 0.866 | 86.6% |
| 45° | 1.000 | 100.0% |
| 60° | 0.866 | 86.6% |
| 75° | 0.500 | 50.0% |
| Location | Gravity in m/s² | Gravity in ft/s² |
|---|---|---|
| Earth (rounded model default) | 9.81 | 32.17 |
| Moon | 1.62 | 5.31 |
| Mars | 3.73 | 12.24 |
Example Calculations
Example 1: Calculate range
Suppose a catapult launches a projectile at 20 m/s at an angle of 45° on Earth, where g = 9.81 m/s².
The range is about 40.77 m.
Example 2: Calculate initial velocity
Suppose the target is 30 m away, the launch angle is 45°, and gravity is 9.81 m/s².
The required initial velocity is about 17.16 m/s.
FAQ
Why is 45 degrees the best angle for maximum range?
In the basic projectile model, range is proportional to sin(2θ). The largest possible value of sine is 1, which happens when 2θ = 90°. That means θ = 45°. This only applies when launch height and landing height are the same and air resistance is ignored.
Why can two launch angles give the same range?
Complementary angles often give the same range. For example, 30 degrees and 60 degrees have the same sin(2θ) value, so they give the same range if the initial velocity and gravity are the same. The calculator returns the lower angle as the primary answer and the complementary higher angle beside it. At maximum range, both merge at 45 degrees.
Why does the calculator show an impossible angle error?
With nonnegative range and positive speed and gravity, an angle is impossible when Rg/vยฒ exceeds 1. Zero speed and range leave the angle undetermined. In this context, it usually means the entered range is too long for the given initial velocity and gravity. You would need a higher launch speed, a shorter range, or different gravity for a real solution.
