How to derive Terminal Velocity equation using Stokes’ law (step by step)

Here we will work on the derivation of the Terminal Velocity equation or formula using Stokes’ Law.

Derivation of Terminal Velocity Equation

When an object is falling through a fluid, in that case, if we want to analyze its motion (and find out its acceleration, if any) then we need to consider the weight of the object, the upthrust on the object applied by the displaced volume of the fluid, and the viscous drag force caused by the movement of the object in the fluid.

Usually, we consider the equilibrium situation, in which the weight of the object exactly balances the sum of upthrust and drag force. And in this equilibrium situation as the net force on the object is zero, hence the velocity of the object remains constant. This constant velocity is terminal velocity.

This is true for skydivers falling through the fluid air as well as for a ball bearing dropping through a column of oil.

To derive the Terminal Velocity equation we will consider simple situations, say for a solid sphere moving slowly in a fluid.

Now in equilibrium, i.e. when the solid sphere is moving with terminal velocity then:

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Thus using Stokes’ law you can derive the terminal velocity equation.

Originally published at https://physicsteacher.in on September 30, 2020.