By Dan Tyler

# Derivation of the acceleration of an object that is
flying circles within a moving air mass

Assume that the object is flying
around the center of a moving circle with an angular velocity of _{}. Assume that
the circle has a radius of _{} and that the
center of the circle is moving with a speed of _{} in the _{} direction. Then, the components of the position
vector in the ground fixed coordinate system are given by:

_{}

Take individual derivatives of the position vector
components to derive the velocity components. You get the following:

_{}

Then, take derivatives again to get the components of the
acceleration vector:

_{}

It is important to note that both components of the
acceleration vector are purely sinusoidal. This is the key to the final result.

The magnitude of the acceleration
vector, _{}, is determined by utilizing the Pythagorean theorem:

_{}

When the acceleration components are plugged into this
expression we find that things simplify nicely:

_{}

The reason for this simplification is that, for any angle _{}, the sum of the squares of the sine and cosine functions is
unity: _{}. For those
comfortable with the notion of centripetal acceleration being the square of the
speed around the center of rotation divided by the radius of rotation, you may
see that this is IDENTICAL, since the angular speed is _{}.

The angle (or direction) of the
acceleration vector is found by using the inverse tangent function as follow:

_{}

The minus signs have not been cancelled because they are
important for determining the quadrant in which the acceleration vector
resides. However, a simple
trigonometric identity for both the sine and cosine functions allows one to
write an equivalent expression that allows us to remove them. The expression is:

_{}

Thus, the angle of the acceleration vector is _{} radians (180^{o})
ahead of the angle of the position vector (the opposite direction), but rotates
at the same rate.

The following image shows graphically
what I derived above. The green
line segments are the radius of the moving circle in relation to the point at
which the object is at time _{}. Note that the
acceleration vector falls exactly on these line segments. Thus, there is NO DIFFERENCE if the
plane is moving within an air mass moving across the surface.