Projectile Motion Range Formula
Calculate the range of a projectile using R = v²sin(2θ)/g.
Learn the ideal projectile motion formula with worked examples.
The Formula
The projectile range formula calculates how far an object travels horizontally when launched at an angle. It assumes ideal conditions: no air resistance, flat ground, and the launch and landing heights are the same.
This is one of the most practical equations in classical mechanics. It applies to everything from a football kick to a cannonball trajectory.
The key insight is that maximum range occurs at a 45° launch angle. Any angle above or below 45° produces a shorter range, and complementary angles (like 30° and 60°) give the same range.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| R | Range — the total horizontal distance traveled | m |
| v₀ | Initial launch speed | m/s |
| θ | Launch angle measured from the horizontal | degrees or radians |
| g | Acceleration due to gravity (9.81 m/s² on Earth) | m/s² |
Derivation Summary
The formula comes from splitting the motion into horizontal and vertical components:
- Horizontal: x = v₀ cos θ × t
- Vertical: y = v₀ sin θ × t − ½gt²
Setting y = 0 to find the time of flight gives t = 2v₀ sin θ / g. Substituting this into the horizontal equation and using the identity 2 sin θ cos θ = sin 2θ yields the range formula.
Example 1
A soccer player kicks a ball at 25 m/s at a 40° angle. How far does it travel?
Identify values: v₀ = 25 m/s, θ = 40°, g = 9.81 m/s²
Calculate sin 2θ: sin(80°) = 0.9848
R = (25² × 0.9848) / 9.81 = (625 × 0.9848) / 9.81
R = 615.5 / 9.81
R ≈ 62.7 m
Example 2
A golfer hits a ball at 70 m/s at exactly 45° (maximum range). How far does it go?
Identify values: v₀ = 70 m/s, θ = 45°, g = 9.81 m/s²
Calculate sin 2θ: sin(90°) = 1.0
R = (70² × 1.0) / 9.81 = 4900 / 9.81
R ≈ 499.5 m (about half a kilometer — this is the theoretical maximum with no air resistance)
When to Use It
- Calculating the range of a thrown or launched object in physics problems
- Determining the optimal launch angle for maximum distance
- Sports science — analyzing kick, throw, or hit distances
- Engineering applications involving ballistic trajectories
- Quick estimates when air resistance is negligible