The Ballistic Flight Parabolic Equation calculator computes the parabolic equation coefficients based on the launch angle above the horizon (θ) at an initial velocity (V) assuming a constant downward acceleration (g).

INSTRUCTIONS:  Choose units and enter the following:

• (V) Initial Launch Velocity
• (θ)  Launch Angle above the horizon.
• (h) Initial Height above the Plane
• (g)  Acceleration due to gravity (default is 9.80665 m/s²)

Parabolic Flight Equation:  The calculator returns the equation of the parabola to match the flight.  

The Math / Science

The formula for the parabolic flight equation is:

     y = a•x² + b•x + c

where:

• a = -4 • M / R² 
• b = 4 • M / R
• c = h (initial height)
• M is the Ballistic Maximum Altitude without h
• R is the Ballistic Maximum Range without h

One of the most common representations of a parabola in nature is an object moving in the gravitational field of a massive body, such as the projectile motion of a body affected by the Earth's gravity.  The figure shows the parabolic trajectory typifying a projectile that is affected solely by gravity, 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

Parabola Calculators

• Parabola Formula:  This computes the y coordinate of a parabola in the form y = a•x²+b•x+c
• Parabolic Area: This computes the area within a section of a parabola
• Parabolic Area (Concave): This computes the outer area of a section of a parabola.
• Parabolic Arc Length: This computes the length a long a segment of a parabola.
• Paraboloid Volume: This is the volume of a parabola rotated around an axis (i.e. paraboloid)
• Paraboloid Surface Area : This is the surface area of a paraboloid.
• Paraboloid Weight: This is the weight or mass of a paraboloid.
• Ballistic Flight Parabolic Equation: This provides the formula of the parabola that matches a ballistic flight.

References

[Figure] Initial velocity of parabolic throwing
Source: Wikipedia / Fizped (modified to include motion equation)
URL: http://en.wikipedia.org/wiki/Projectile_motion#mediaviewer/File:Ferde_hajitas1.svg