Derive an expression for terminal velocity acquired by a spherical body falling in viscous media under gravity.

Let during fall, v be the velocity of body at any instant.
The different forces acting on the body at that instant are:

(i) Weight W, acting downward,
(ii) Upward thrust U, acting upward,
(iii) Viscous force F, acting upward.
Net force on body is,  
      
We can see that the body falls under gravity.
Therefore, its velocity will increase due to gravity.
But as the velocity increases, the viscous force (factor II) acting on the body also increase.
Since factor I is constant, therefore the net force on the body decreases with time.
At some velocity v = v_o, the resultant of all the forces acting on body reduces to zero and body acquires the constant velocity known as terminal velocity.
Therefore, 

The above expression represents the terminal velocity acquired by a spherical body falling in viscous media under gravity.