# Airplane Angle Related Rate

- Author:
- Dr. Doug Davis, 3D

This is a related rate problem. The airplane is flying at a constant speed and altitude toward a point. The question is how fast is the view angle increasing as the plane flies closer. Note: the airplane may not appear in some browsers.
THE MATH
The math is simpler in Radians so find in radians per second, then convert to per second.
The primary relation is where is the altitude and is the distance in kft ( 1000 feet).
The speed is in Miles per Hour which is converted to Velocity by multiplying by
Taking the derivative with respect to time of the tangent relation above gives
and
Also from Pythagorean Theorem
A little algebra then gives
Then converting to degrees

As the plane flies closer, how does the rate of change of the angle change?
Is the rate the same at any altitude?
How does the rate of change of angle vary with the airplane's speed?