Terminal velocity is the maximum constant velocity acquired by the body while falling freely in a viscous medium.

When a small spherical body falls freely through a viscous medium, three forces act on it

• Weight of the body acting vertically downwards
• Upward thrust equal to the amount of water displaced
• Viscosity acting in a direction opposite to the direction of motion

1.  The weight of the body acting vertically downwards = W= mg = Volume * Density * g = 4/3 ∏ r³ ρ g
2. Upward thrust due to buoyancy = weight of the medium displaced = mg = Volume * Density of the medium displaced * g = 4/3 ∏ r³ σ g
3. The drag due to viscous factor acts in a direction opposite to the motion of the body. According to Stoke’s law F is proportional to v or F = 6 ∏ π rv

Resolving the above three forces when the body reaches the terminal velocity is given by

Upward thrust due to buoyancy + Viscous drag = Weight of the body which is F_T + F_V = W

4/3 ∏ r³ σ g + 6 ∏ π rv = 4/3 ∏ r³ ρ g

 Solving the above equation, gives the terminial velocity as v = 2 r² (ρ – σ) g / 9 π