Formula & Calculator
Projectile Range
Calculates the horizontal distance a projectile travels before landing, assuming it launches and lands at the same height with no air resistance.
Interpretation
Projectile range: R = v²·sin(2θ)/g, where v is initial speed, θ is launch angle, g is gravity. Gives horizontal distance covered. Example: v=20 m/s, θ=45° → R = 400×1/9.81 ≈ 40.8 m.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| R | Horizontal range | m |
| v | Initial launch speed | m/s |
| theta | Launch angle above horizontal | degrees |
| g | Gravitational acceleration | 9.81 m/s2 |
What it means
The range of a projectile is the horizontal distance it travels from launch to impact, assuming it lands at the same height. The formula R = v² sin(2θ)/g is derived from kinematics and assumes no air resistance. The range is maximum when θ = 45°. This equation is used in sports (javelin, shot put), military ballistics, and satellite launch calculations. It is also used in engineering for designing trajectories of projectiles and in physics to study motion under gravity. The formula highlights the influence of initial speed and launch angle. In practice, air resistance reduces the range, and corrections are made using more complex models. Understanding projectile motion is fundamental for analyzing motion in two dimensions.
Worked example
Projectile Range – Two Examples
Real‑World| Parameter | Value |
|---|---|
| v | 20 m/s |
| θ | 45° |
| Parameter | Value |
|---|---|
| v | 30 m/s |
| θ | 30° |
Common mistakes
- Angle θ: The launch angle with respect to the horizontal – not the vertical.
- Initial speed v: The magnitude of the launch velocity – not a component.
- Range R: Horizontal displacement – only valid if landing height equals launch height.
- Units: v in m/s, θ in radians/degrees (sine function uses degrees if calculator set), g in m/s² → R in m.
- Air resistance: This formula neglects air resistance; in reality, range is reduced.
Applications
The range of a projectile, R = v² sin(2θ)/g, gives the horizontal distance travelled by a projectile launched at speed v and angle θ. This formula is essential in ballistics, sports, and military engineering. It is used to calculate the optimal launch angle (45° in vacuum) for maximum range, and to design artillery trajectories and golf drives. In civil engineering, it is applied to the design of water jets and sprinklers. In aerospace, it helps with re‑entry trajectories and landing site prediction. By using this formula, engineers can predict where a projectile will land, which is crucial for safety, accuracy, and efficiency in many practical applications.
- Ballistic trajectory design for artillery and missiles
- Sports performance analysis (golf, javelin, long jump)
- Design of irrigation sprinklers and water jets
- Re‑entry trajectory prediction for spacecraft
- Launch angle optimisation for maximum distance
Frequently Asked Questions
The horizontal range (distance travelled) for a projectile launched with initial speed v at angle θ above the horizontal is R = v²·sin(2θ)/g. This assumes the launch and landing heights are equal, and air resistance is neglected.
Applying it when the launch and landing heights are different (e.g., launched from a cliff). In that case, you must use the full trajectory equations.
The maximum range is achieved at θ = 45° (since sin(90°) = 1). For any other angle, the range is smaller.
Range is proportional to v². Doubling the launch speed quadruples the range (ignoring air resistance).
Range is inversely proportional to g. On the Moon (lower g), the range is larger for the same launch conditions.
R = v²·sin(60°)/g = v²·(√3/2)/g. This is about 86.6% of the maximum range (v²/g).
In the absence of air resistance, the range is independent of mass. With air resistance, heavier projectiles tend to have longer ranges.
Use the equations: t = (v·sinθ + √(v²sin²θ + 2gh))/g (positive root), then R = v·cosθ · t. The formula is more complex than the flat‑ground case.