An object which is falling through the atmosphere is subjected to two external forces. One force is the gravitational force, expressed as the weight of the object. The other force is the air resistance, or drag of the object. The motion of a falling object can be described by Newton's second law of motion (Force equals mass times acceleration -- F = m a) which can be solved for the acceleration of the object in terms of the net external force and the mass of the object.

a = F / m

The net external force (F) is equal to the difference between the
weight and the drag forces (W - D). When drag is equal to weight, there is no
net external force on the object and the object will fall at a
constant velocity as described by Newton's first
law of motion. The constant velocity is called the ** terminal
velocity **.

We can determine the value of the terminal velocity by doing a little algebra and using the drag equation. Drag (D) depends on a drag coefficient, (Cd) the air density, (r) the square of the air velocity (V) and some reference area (A) of the object.

D = Cd * r * V ^2 * A / 2

At terminal velocity, D = W. Solving for the velocity, we obtain the equation

V = sqrt ( (2 * W) / (Cd * r * A) )

Typical values of the drag coefficient are given on a separate slide.

The terminal velocity equation tells us that an object with a
large cross-sectional area or a high drag coefficient will fall
slower than an object with a small area or low drag coefficient. (A
large flat plate will fall slower than an a small ball with the same
weight.) And if we had two objects with the same area and drag
coefficient (two identically sized spheres), the lighter object would
fall slower. This seems to contradict the findings of Galileo that
all free falling objects would fall at the
same rate with equal air resistance. But Galileo's principal only
applies in a vacuum, where there is **NO** air resistance and drag
is equal to zero.

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*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *