# Ballistic Base Elevation

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The Ballistic Base Elevation calculator computes the base height (elevation) needed above the horizon (h) for a launch point to achieve a maximum range (horizontal distance) traveled by an object based on the ballistic flight initial velocity (V) of the object, and angle of launch (θ), and the vertical acceleration (g).

INSTRUCTIONS: Choose units and enter the following:

• (r)  This is the range of the ballistic flight (distance down range)
• (V)  This is the initial velocity of the object that is launched (e.g. muzzle velocity from the gun or cannon)
• (θ)  This is the launch angle above the horizon.
• (g)  This is the acceleration due to gravity with a default of  9.80665 m/s2

Ballistic Base Elevation (h): The calculator returns the height (elevation) of the base in meters.  However, this can be automatically converted to other distance units (e.g. miles or kilometers) via the pull-down menu.

Related Calculators:

### The Math / Science

The Ballistic Range equation:

R = ((V * sin(theta) + sqrt( (V *sin(theta))^2 + (2*g*h))) /  g) * cos(theta)*V)

where:

• R is the ballistic range
• V is the initial velocity
• theta is the launch angle
• h is the elevation (height) of the base launch point
• g is the acceleration due to gravity.

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).  If we solve for the height, the formula is as follows:

h = ( ( (g*r)/(cos(theta) * V) - V*sin(theta))^2   - (V*sin(theta))^2)/(2*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 drag (see below).  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 ->International Gravity Formula 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.