Understanding Projectile Range
Projectile range is the horizontal distance an object covers from the moment it leaves the launch point until it returns to its landing height. Unlike vertical motion, gravity produces no horizontal acceleration—the object maintains constant horizontal velocity throughout flight. This separation of horizontal and vertical motion is what makes range calculations tractable.
Range depends on three factors:
- Initial velocity: Higher launch speeds produce greater distances.
- Launch angle: The angle between the velocity vector and the ground critically affects range. A 45° angle maximises distance on level ground, though this changes with elevation differences.
- Initial height: Launching from an elevated position extends flight time and therefore range, even at lower angles.
Notably, mass is irrelevant. A tennis ball and a cannonball with identical launch conditions travel the same distance (ignoring air resistance).
Range Formulas for Projectile Motion
Projectile motion splits into two scenarios depending on whether you launch from ground level or from an elevated position.
From ground level (h₀ = 0):
x = (v² × sin(2α)) / g
x— Horizontal range (distance in metres)v— Initial velocity magnitude (metres per second)α— Launch angle in degreesg— Gravitational acceleration (9.81 m/s²)
Range with Initial Height
When launching from height h₀ above the landing surface, you must first find flight time, then apply the range equation.
t = (v×sin(α) + √[(v×sin(α))² + 2gh₀]) / g
x = v×cos(α) × t
t— Time of flight in secondsv— Initial velocity (m/s)α— Launch angle in degreesh₀— Initial height above landing surface (metres)g— Gravitational acceleration (9.81 m/s²)x— Horizontal range (metres)
The Optimal Launch Angle
On level ground with no initial height, the 45° angle maximises range. At this angle, sin(2 × 45°) = sin(90°) = 1, making the numerator as large as possible.
However, this changes when launching from height. Higher launch points shift the optimal angle slightly downward from 45°. The gain in flight time from elevation allows shallower angles to outperform 45°. For example, throwing from a 2 m height with moderate velocity shows optimal range closer to 40° or less.
Conversely, launching at 90° (straight up) yields zero range—the projectile returns to the starting point. A 0° launch (horizontal throw) also produces limited range on level ground but extends significantly from elevated positions.
Common Pitfalls and Considerations
Avoid these frequent mistakes when calculating projectile range:
- Forgetting to convert angles to radians — Most calculators default to degrees for user input but require radians internally. Always verify your tool's convention. The relationship is: radians = degrees × (π/180).
- Neglecting air resistance — These formulas assume a vacuum or negligible drag. Real projectiles experience air resistance that reduces range significantly, especially for light objects or high velocities. Wind, spin, and shape all matter in practice.
- Confusing initial height with landing height — The height h₀ is measured from the launch point. If you throw from a 10 m cliff and the projectile lands 5 m below, use h₀ = 5 m, not 10 m. Negative landing heights are invalid in this formula.
- Using wrong gravity value — This calculator assumes g = 9.81 m/s² (standard Earth). On the Moon (1.62 m/s²) or Jupiter (24.79 m/s²), ranges differ dramatically. Always match your gravity constant to the location.