Calculate catapult range, initial velocity, launch angle, or gravity from any three projectile motion values in your chosen units.
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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. Enter any three values to calculate the missing value.
R = \frac{v^2 \sin(2\theta)}{g}v = \sqrt{\frac{R g}{\sin(2\theta)}}\theta = \frac{1}{2}\sin^{-1}\left(\frac{R g}{v^2}\right)g = \frac{v^2 \sin(2\theta)}{R}- 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 uses the inverse sine relationship. 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, standard | 9.81 | 32.17 |
| Moon | 1.62 | 5.31 |
| Mars | 3.71 | 12.17 |
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.
R = \frac{20^2 \sin(2 \cdot 45^\circ)}{9.81}R = \frac{400 \cdot 1}{9.81} = 40.77\text{ m}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.
v = \sqrt{\frac{30 \cdot 9.81}{\sin(2 \cdot 45^\circ)}}v = \sqrt{294.3} = 17.15\text{ m/s}The required initial velocity is about 17.15 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 angle-solving formula returns the lower angle from the inverse sine result.
Why does the calculator show an impossible angle error?
An angle is impossible when the value inside the inverse sine is less than -1 or greater than 1. 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.