How To Calculate Terminal Velocity

Terminal velocity describes the equilibrium point in kinematics where atmospheric drag on a falling object becomes equal and opposite to the acceleration due to gravity. Thus, the object cannot accelerate further without outside assistance, and has reached its highest possible velocity in that medium.

Drag is a function of the aerodynamics of the object in question: an umbrella would fall much more slowly than a missile of the same weight. We can use the terminal velocity equation to calculate an object's speed at this point.

Step 1

Determine the weight ​W​ of the falling object. The easiest way to do this is usually to measure this quantity directly. You can also estimate weight if you know the construction materials and dimensions.

Step 2

Calculate the frontal area ​A​ of the falling object. The frontal area is the apparent area facing in the direction of falling. You can determine this area by measuring the outline of the object from that orientation.

For example, if the falling object were a cone, the tip of the cone would point straight downward, and the frontal area would appear to be a circle equal to the area of the circular base of the cone.

Step 3

Determine the drag coefficient ​C​_d of the falling object. You can usually avoid having to calculate the drag coefficient yourself by looking up an approximate value in a reference book or on the Internet. If you need a highly precise value, you should consult with an engineer.

Step 4

Determine the atmospheric density ​ρ ​of the medium through which the object will be falling. If the medium is the air, then you should know that air density decreases with altitude, which means that the object's terminal velocity will decrease as it gets closer to the ground (where the gas is denser and pushes back harder, providing stronger braking power).

Thus you can calculate terminal velocity at any one altitude using simple mathematics, but to calculate the change in the terminal velocity over a long-distance fall, you will require the use of calculus or empirical approximations.

Air density also changes with the weather; there is no uniform density value for a given altitude. To get the most accurate measurements of air density, you will need to multiply average air density values by local weather condition offsets.

Atmospheric information is available in the United States from the National Weather Service, a service of the National Oceanic and Atmospheric Administration.

Step 5

For any given altitude, the terminal velocity equation is:

\(V_t=\sqrt{\frac{2W}{\rho A C_d}}\)

where ​W​ is is the object's weight, ​ρ​ is the density of the gas, ​A​ is the cross sectional area of the object, and ​C​_d is the drag coefficient.

In plain English, the terminal velocity of the object is equal to the square root of the quotient of twice the object's weight over the product of the object's frontal area, its drag coefficient, and the gas density of the medium through which the object is falling.