Projectile Range Calculator Formula
Understand the math behind the projectile range calculator. Each variable explained with a worked example.
Formulas Used
Horizontal Range
range = pow(velocity, 2) * sin(2 * angle_rad) / gravityVariables
| Variable | Description | Default |
|---|---|---|
velocity | Launch Velocity(m/s) | 30 |
angle | Launch Angle(degrees) | 45 |
gravity | Gravitational Acceleration(m/s²) | 9.81 |
angle_rad | Derived value= angle * pi / 180 | calculated |
How It Works
Projectile Range
For a projectile launched from and landing at the same elevation:
Formula
R = v² * sin(2*theta) / g
Maximum range occurs at a 45-degree launch angle. The range is symmetric about 45 degrees (e.g., 30 and 60 degrees give the same range).
Worked Example
A ball is launched at 30 m/s at 45 degrees.
- 01R = v² * sin(2 * theta) / g
- 02R = 900 * sin(90°) / 9.81
- 03R = 900 * 1 / 9.81
- 04R = 91.74 m
Frequently Asked Questions
At what angle is the range maximized?
45 degrees gives the maximum range on flat ground, because sin(2 * 45°) = sin(90°) = 1.
Does air resistance affect projectile range?
Yes. This formula assumes no air resistance. In reality, drag reduces both range and maximum height.
Why do complementary angles give the same range?
Because sin(2*theta) = sin(180° - 2*theta). For example, sin(60°) = sin(120°), so 30° and 60° produce equal ranges.
Learn More
Guide
How to Calculate Velocity and Acceleration
Learn how to calculate velocity and acceleration step by step. Covers average and instantaneous velocity, uniform acceleration, kinematic equations, and worked examples.
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Open Projectile Range Calculator