Terminal Velocity

Any fluid exists in layered form. The friction between the layers of the fluid causes drag. This drag which affects the motion of the fluid is termed as viscosity. When an object is dropped in a fluid, the viscosity of the fluid affects the motion of the object. During its motion the highest velocity attained by the object when dropped in a particular fluid is termed as terminal velocity. The terminal velocity of the object in the fluid is observed when the sum total of the drag and the buoyancy is equal to the net gravitational force exerting on the object. As the forces acting on the object are equal in magnitude but opposite in direction, they cancel out each other, hence, the acceleration of the object becomes zero.

Mathematical expression of the terminal velocity

In order to attain the terminal velocity while moving in the fluid, the object should move with the constant speed inside the fluid countering the opposite force exerted by the fluid.

vt= 2mgρACd 

here, vt denotes terminal velocity, g is acceleration due to gravity, Cd denotes coefficient of drag, denotes density of the fluid in which the object is dropped and A denotes the surface area of the object dropped.

This mathematical expression of the terminal velocity can be obtained as follows:

F=bv2 

here, b denotes constant describing the drag in the fluid.

When an object falls freely in the fluid. sum total of all the forces F is

F=ma 

Here, m is the mass of the object and a denotes the acceleration during free fall.

Assume that the fall of the object is happening in positive direction,

• mg-bv2=ma
• mg-bv2=mdvdt (acceleration is rate of change of velocity)
• dtmg-bv2=mdv
• dtmmg-bv2=dv

Integrating both sides, we get

• 1mdt= dvmg-bv2
• dvmg-bv2=1bdv2- v2

here, α= mgb

Put v= αtanhϑ

• dv = αSech2ϑ dϑ
• v2= 2tan2hϑ

Upon integrating we get,

1bSech2ϑ dϑ2- 2tan2hϑ

• 1bSech2ϑ dϑ2(1- tan2hϑ)
• 1bSech2ϑ dϑ2(Sec2hϑ) = 1αbdϑ = 1αbarctanh v+C , where C denotes the constant of integration
• 1mt=1αbarctanh v+C
• vt=αtanh (αbm t +arctanh v0)  

By substituting α= mgb in vt=αtanh tbgm+arc tanhv0

• vt=mgbtanh tbgm+arc tanhv0
• when the limit t→∞, we get vt= 2mgρACd, where vt is the terminal velocity.