Horizontal Range
Simple Explanation
The horizontal range is the total horizontal distance a projectile covers before landing back at its launch height. It depends on launch speed, launch angle, and gravity.
Why Do We Need It?
Range is often the most practically important measure of a projectile's motion — how far a thrown object, a kicked ball, or a launched object actually travels.
Formula
Horizontal Range (Projectile Motion)
R = u² sin2θ / g
The total horizontal distance a projectile travels before landing back at its launch height.
- R
- — Horizontal range, in metres
- u
- — Initial launch speed, in m/s
- θ
- — Launch angle above the horizontal
- g
- — Acceleration due to gravity, 9.8 m/s²
When to use it: Use to find how far a projectile travels horizontally — note the range is maximised when θ = 45°, since sin2θ is then at its maximum value of 1.
Worked Example
Finding horizontal range, and the angle for maximum range
A cannonball is launched at 40 m/s at 45° above horizontal. Find its range (g = 9.8 m/s², sin90° = 1).
Why Does This Work?
Range is the product of the (constant) horizontal velocity and the total time of flight — combining those two relationships and simplifying with a trigonometric identity produces the sin2θ term, which is exactly why range is maximised at a 45° launch angle (where sin2θ = sin90° = 1, its largest possible value).
Real-Life Example
Why long jumpers and shot-putters aim for roughly 45°
Athletes in throwing and jumping events are coached to launch at an angle well below 90°, often closer to 45°.
Since range is maximised at a 45° launch angle for a given speed (ignoring air resistance and the launch height above the ground), athletes aiming for maximum horizontal distance train toward launch angles in that range rather than launching as steeply as possible.
Practice
A projectile is launched at 20 m/s at 30° above horizontal. Find its range (g = 9.8 m/s², sin60° ≈ 0.866).
HardCommon mistake
Forgetting that the angle inside the sine is DOUBLE the launch angle (sin2θ, not sinθ) — this is easy to miss and produces a very different (usually much smaller) incorrect answer.
Quick Review
- R = u² sin2θ / g gives the horizontal distance travelled.
- Range is maximised at a 45° launch angle, where sin2θ = 1.
- The same range is achieved by complementary angles (e.g. 30° and 60°) for a given speed.