Ballistic Range

The Max Ballistic Range calculator computes the maximum range (horizontal distance) traveled by an object based on the initial velocity (V) of the object, an angle of launch (θ), the launch point's height (h) above the plane, and the acceleration due to gravity (g).

INSTRUCTIONS: Choose units and enter the following:

• (V)  Initial Velocity (e.g. muzzle velocity from the gun or cannon)
• (θ)  Launch Angle above the horizon.
• (h)  Initial Height of the launch point above the plane.
• (g)  Acceleration due to gravity with a default of  9.80665 m/s² 

Max Range (x): The calculator returns the maximum distance down range (x) in meters.  However, this can be automatically converted to other distance units (e.g. miles or kilometers) via the pull-down menu.

The Math / Science

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.   

Gravity Calculators:

• Acceleration Due to Gravity
• Acceleration Due to Gravity based on Latitude
• Acceleration Due to Gravity based on Altitude
• Acceleration Due to Gravity on Earth at Sea Level
• Force of Gravity between two Objects
• Universal Gravity Constant (G)

Ballistic Flight Calculators:

• Ballistic Maximum Altitude:  This is the maximum altitude achieved in free ballistic flight.
• Ballistic Maximum Range:  This is the maximum horizontal range.
• Ballistic Flight Time:  This is the time duration of free flight.
• Simplified Ballistic Range: This is the range with no initial elevation above the plane.
• Ballistic Vertical Velocity: This is the vertical velocity at a given time.
• Ballistic Horizontal Velocity: This is horizontal velocity or ground speed.
• Vertical Position (Y) in Ballistic Flight: This compute the vertical position (y) at a given time within ballistic flight.
• Horizontal Position (X) in Ballistic Flight: This compute the horizontal position (x) at a given time within ballistic flight..
• Ballistic Position at Time (t): This compute the position (x,y) at a given time within ballistic flight, where x is distance down range and y is the height above the plane.
• Ballistic Parabolic Equation provides the parabolic flight position equation based on the launch speed, height and angle.
• Acceleration Due to Gravity at Sea Level
• Velocity to achieve a Max Ballistic Height: This computes the initial velocity required to achieve the max height.
• International Gravity Equation
• Force of Earth's Gravity
• Force of Drag