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State & explain terminal velocity. ​

Answer 1

When a body is dropped in a viscous fluid, it is first accelerated and then its acceleration becomes zero and it attains a constant velocity called terminal velocity.

Consider a small ball of radius r, which is gently dropped into the liquid of infinite extent. As the steel ball falls, it experiences three forces: the force of gravity W, the buoyant force $\sf{F_B}$, the drag force $\sf{F_D}$.

The free-body diagram of the ball is shown in the figure. If $\sf{\rho_0}$ be the density of the body and $\sf{\rho}$ be the density of liquid, then

$\sf{W=\dfrac{4}{3}\pi\:r^3\:\rho_0\:g}$

and $\sf{F_B=\dfrac{4}{3}\pi\:r^3\:\rho\:g}$

The magnitude of the drag force is $\sf{F_D=6\pi\:\eta\:r\:v}$.

As the ball falls under gravity, its net weight $\sf{W-F_B}$ is opposed by liquid resistance $\sf{F_D}$. Initially, the drag force is small and as the body gains velocity the drag force also increases at a particular velocity, called the terminal velocity, the magnitude of drag force becomes equal to the effective weight of the body.

At this moment, the net force on the ball becomes zero and it moves with constant velocity. That is,

$\sf{F_D=W-F_B}$

$\sf{6\pi\:\eta\:r\:v=\dfrac{4}{3}\pi\:r^3\left(\rho_0-\rho\right)g}$

$\boxed{\sf{v=\dfrac{2\:r^2\left(\rho_0-\rho\right)g}{9\eta}}}$

This is the terminal velocity.