Computer drawing of a falling ball subject to gravitational and
 drag forces. Terminal velocity = function of weight and drag coefficient.


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
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