Terminal Velocity Derivation

What is Terminal Velocity?

Terminal velocity is defined as the highest velocity attained by an object falling through a fluid. It is observed when the sum of drag force and buoyancy is equal to the downward gravity force acting on the object. The acceleration of the object is zero as the net force acting on the object is zero.

How to find Terminal Velocity?

In fluid mechanics, for an object to attain its terminal velocity, it should have a constant speed against the force exerted by the fluid through which it is moving.

The mathematical representation of terminal velocity is:

\(\begin{array}{l}v_{t}=\sqrt{\frac{2mg}{\rho AC_{d}}}\end{array} \)

Where,

vt is the terminal velocity, m is the mass of the falling object, g is the acceleration due to gravity, Cd is the drag coefficient, 𝜌 density of the fluid through which the object is falling, and A is the area projected by the object.

Terminal Velocity Derivation

Deriving terminal velocity using mathematical terms according to the drag equation as follows:

\(\begin{array}{l}F=bv^{2}\end{array} \)

Where b is the constant depending on the type of drag

\(\begin{array}{l}\sum F=ma\,\, \textup{(free fall of an object)}\end{array} \)

\(\begin{array}{l}mg-bv^{2}=ma\,\,\textup{(assuming that the free fall is happening in positive direction)}\end{array} \)

\(\begin{array}{l}mg-bv^{2}=m\frac{dv}{dt}\end{array} \)

\(\begin{array}{l}\frac{1}{m}dt=\frac{dv}{mg-bv^{2}\,\,\textup{(differential form of the equations)}}\end{array} \)

\(\begin{array}{l}\int \frac{1}{m}dt=\int \frac{dv}{mg-bv^{2}}\end{array} \)
(integrating the equations)

\(\begin{array}{l}\int \frac{dv}{mg-bv^{2}}=\frac{1}{b}\int \frac{dv}{\alpha ^{2}-v^{2}}\end{array} \)

Where,

\(\begin{array}{l}\alpha =\sqrt{\frac{mg}{b}}\end{array} \)

\(\begin{array}{l}dv=\alpha sech^{2}(\Theta )d\Theta\,\, (\textup{after substituting for}\,\, v=\alpha tanh(\Theta ))\end{array} \)

\(\begin{array}{l}v^{2}=\alpha ^{2}tanh^{2}(\Theta )\end{array} \)

After integration,

\(\begin{array}{l}\frac{1}{b}\int \frac{\alpha sech^{2}(\Theta )d\Theta }{\alpha ^{2}-\alpha ^{2}tanh^{2}(\Theta )}\end{array} \)

\(\begin{array}{l}\frac{1}{b}\int \frac{\alpha sech^{2}(\Theta )d\Theta }{\alpha ^{2}(1-tan^{2}\Theta )}\end{array} \)

\(\begin{array}{l}\frac{1}{b}\int \frac{\alpha sech^{2}(\Theta )d\Theta }{\alpha ^{2}sech^{2}(\Theta )}=\frac{1}{\alpha b}\int d\Theta =\frac{1}{\alpha b}arctanh(\frac{v}{\alpha })+C \,\,(\textup{using the identity}\,\, 1-tanh^{2}(\Theta )=sech^{2}(\Theta ) )\end{array} \)

\(\begin{array}{l}\frac{1}{m}t=\frac{1}{\alpha b}arctanh(\frac{v}{\alpha })+C\,\, \textup{(from original equation)}\end{array} \)

\(\begin{array}{l}v(t)=\alpha tanh(\frac{\alpha b}{m}t+arctanh(\frac{v_{0}}{\alpha }))\end{array} \)

\(\begin{array}{l}\textup{By substituting for}\,\, \alpha =\sqrt{\frac{mg}{b}}\end{array} \)
\(\begin{array}{l}v(t)=\alpha tanh(t\sqrt{\frac{bg}{m}}+arctanh(\frac{v_{0}}{\alpha }))\end{array} \)

\(\begin{array}{l}v(t)=\sqrt{\frac{mg}{b}}tanh(t\sqrt{\frac{bg}{m}}+arctanh(v_{0}\sqrt{\frac{b}{mg}}))\end{array} \)

After substituting for vt

\(\begin{array}{l}\lim_{t\rightarrow \infty }v(t)=\lim_{t\rightarrow \infty }(\sqrt{\frac{mg}{b}}tanh(t\sqrt{\frac{bg}{m}}+arctanh(v_{0}\sqrt{\frac{b}{mg}})))=\sqrt{\frac{mg}{b}}\end{array} \)

\(\begin{array}{l}∴ v_{t}=\sqrt{\frac{2mg}{\rho AC_{d}}}\end{array} \)

Therefore, above is the derivation of terminal velocity.

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Frequently Asked Questions – FAQs

When does terminal velocity exist?

When the speed of a moving object is no longer increasing or decreasing; the object’s acceleration (or deceleration) is zero.

What is terminal velocity?

Terminal velocity is defined as the highest velocity attained by an object that is falling through a fluid.

Who discovered terminal velocity?

Galileo discovered terminal velocity.

Does terminal velocity exist in a vacuum?

In vacuum since there is no drag force, the terminal velocity does not exist.

How does terminal velocity work?

Terminal velocity, steady speed achieved by an object freely falling through a gas or liquid. An object dropped from rest will increase its speed until it reaches terminal velocity; an object forced to move faster than its terminal velocity will, upon release, slow down to this constant velocity.

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