The Bullet Flight Range calculator computes the maximum range (horizontal distance) traveled by a bullet based on the muzzle velocity (V), elevation angle (α), shooter height (h), and the acceleration due to gravity. Note, this does not account for air resistance or wind factors.
INSTRUCTIONS: Choose units and enter the following:
- (V) Muzzle Velocity
- (α) Elevation Angle above the horizon.
- (h) Shooter Height
Max Bullet Range (R): The calculator returns the maximum distance down range in feet. However, this can be automatically converted to other distance units (e.g. miles or meters) via the pull-down menu.
The Math / Science
The range of a bullet can be estimated using basic physics. Once the bullet is fired at a specific muzzle velocity, gravity immediately starts to pull the bullet down toward the center of the Earth. Wind resistance slows the bullet, and wind direction will put it off target. This calculator only accounts for the pull of gravity, but it does account for the elevation angle and the shooter height, since both, along with muzzle velocity and the force of gravity, pay a major impact in the flight of the bullet.
The Maximum Ballistic Range equation is:
`R = ((V • sinθ + sqrt( (V •sinθ)² + (2•g•h))) / g) • cosθ•V`
where:
- R is the max ballistic range
- V is the initial velocity
- θ is the launch angle
- h is the initial height or elevation above the plane
- g is the acceleration due to gravity
This formula algebraically equivalent in the following form:
`R = V^2/(2g) * ( 1 + sqrt(1 + (2gh)/(v^2sin^2θ)))*sin2θ`
The Ballistic Range equation calculates the horizontal displacement (distance) of an object in free flight. It only takes into account the initial velocity and launch angle (also knows as the loft) and the effects of gravity through an acceleration towards the ground. This formula does not take into account other factors such as the force of drag. A default is provided for the acceleration due to gravity of 9.80665 m/s2 which is mean acceleration (at all latitudes) for sea level on Earth.
Acceleration Due to gravity
The force of gravity pulls masses towards each other. In the case of small objects (e.g. you, an arrow or the Space Shuttle) verses planetary objects (e.g. the Earth or Moon), the difference in masses results in a negligible acceleration of the large object toward the small. Acceleration due to gravity changes based on the mass of the attracting object (e.g. the Earth 9.8 m/s2 verses the moon1.6 m/s2) and the distance from the center of mass. For example, since the Earth is not a perfect sphere, and more closely represented as an oblate spheroid, acceleration due to Earth gravity as Sea Level is more accurately calculated based on latitude. The international gravity formula provides an acceleration due to gravity based on latitude.
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