Projectile Range Calculator
The horizontal range of a projectile launched from ground level at a given speed and angle can be calculated with the formula R = v² sin(2θ) / g. This assumes level terrain, no air resistance, and constant gravitational acceleration. Enter the initial speed (in m/s), the launch angle in degrees, and the gravitational acceleration (9.81 m/s² at Earth's surface) to find the horizontal distance the projectile will travel before returning to the launch height. The range is maximized at 45 degrees, and complementary angles (such as 30° and 60°) produce identical ranges.
Projectile range formula
R = v² × sin(2θ) / g
Where v is the initial speed (m/s), θ is the launch angle in degrees, and g is gravitational acceleration (9.81 m/s²). The time of flight is T = 2v sin(θ) / g.
Understanding projectile range
- The formula applies only when the projectile lands at the same height from which it was launched.
- Complementary angles (for example 30° and 60°) yield identical ranges because sin(2 x 30°) = sin(60°) = sin(120°) = sin(2 x 60°).
- Range increases with the square of velocity: doubling the speed quadruples the range.
- At 0° or 90° the range is zero: the object either stays on the ground or goes straight up and comes back down.
- On the Moon (g = 1.62 m/s²) the same projectile travels about six times farther.
Projectile range: frequently asked questions
What is the range of a projectile?
The range is the horizontal distance a projectile travels from launch to landing, assuming it returns to the same height. It depends on initial speed, launch angle, and gravitational acceleration.
Which angle gives maximum range?
A launch angle of 45 degrees gives the maximum range on level ground, because sin(90°) = 1, which is the maximum value of sin(2θ). Any angle above or below 45° produces a shorter range.
What value of g should I use?
The standard gravitational acceleration at Earth's surface is 9.81 m/s² (or 9.80665 m/s² precisely per NIST). The calculator uses 9.81 m/s² by default. On other planets you would enter the appropriate value.
Does this formula account for air resistance?
No. The formula R = v² sin(2θ)/g assumes a vacuum, meaning no air resistance and no wind. Real-world projectiles will travel shorter distances due to drag. For low-speed objects this estimate is useful; for high-speed or light objects it over-estimates range.
Can I use this for different units?
Yes, as long as you are consistent. If speed is in m/s and g is in m/s², range is in meters. If speed is in ft/s and g is in ft/s² (32.174 ft/s²), range is in feet. Do not mix unit systems.
Official sources
- NIST Reference on Constants, Units, and Uncertainty: Standard acceleration of gravity.
- OpenStax University Physics Volume 1: Projectile Motion.
- NASA Glenn Research Center: Range of a Projectile.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.