Projectile Motion Range Formula
Reference for ideal projectile motion.
Calculate range, max height, and flight time using R = v squared times sin(2theta)/g with angle and velocity 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