Cannon Range

The Cannon Ball Range calculator computes the maximum range (horizontal distance) achieve by a projectile based on the muzzle velocity, angle of fire, cannon height above the plane, and the acceleration due to gravity (g). 

INSTRUCTIONS: Choose units and enter the following:

• (V)  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 

Cannon Ball 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`

This formula calculates the range (horizontal distance) traveled by an object based on the  height (h) above the horizon of the launch point, initial velocity (V) of the object, and angle of launch (theta), and the vertical acceleration (g). 

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 result in a negligible acceleration of the large object toward the small and small object accelerating toward the center of mass of the large object.  Acceleration due to gravity changes based on the mass of the 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: click here -> The international gravity formula provide an acceleration due to gravity based on latitude.  CLICK HERE for the acceleration due to gravity for the other planets in the solar system. 

Ballistic Flight Equations and 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.
• Ballistic Position: This computes the X,Y position at a given time.
• Ballistic Vertical Velocity: This is the vertical velocity at a given time.
• Ballistic Horizontal Velocity: This is horizontal velocity or ground speed.
• Vertical Position in Ballistic Flight: This compute the vertical position (y) at a given time within ballistic flight.
• Horizontal Position in Ballistic Flight: This compute the horizontal position (x) at a given time within ballistic flight.
• Velocity to achieve a Max Ballistic Height: This computes the initial velocity required to achieve the max height.

• Travel Time
• International Gravity Equation
• Force of Earth's Gravity
• Force of Drag