How to solve Numerical problems on terminal velocity

In this post, we will see how to solve numerical problems on terminal velocity. We will discuss the concept in brief and then take sample problem/s and solve them using the appropriate formula of terminal velocity.

Theory & formula to solve terminal velocity numerical

The first numerical problem in this post is about a spherical ball that falls through a fluid. When a spherical object or sphere is released and allowed to fall freely in a fluid, three forces act on it: its weight, W, the upthrust, U, and the viscous drag, F. Remember that U and F act upwards and W acts downwards.

Initially, a net resultant force, R = W – (U + F) will make the sphere fall downward with an acceleration. According to Stokes’ law, viscous drag is directly proportional to the velocity of the falling object.

Hence, in this case, as the velocity of the sphere increases, the viscous drag also increases according to Stokes’ law until (U + F) = W. The resultant force R then becomes zero, and the sphere continues to fall at a constant velocity known as the terminal velocity.

So the formula we are going to use can be simplified like this:
(U + F) = W
=> F = W – U
=> 6πη r ν = W – U. [ Drag force F = 6πη r ν as per Stoke’ law, where η = viscosity, r = radius of the sphere, and v is the terminal velocity]

Numerical problem on terminal velocity with solution

Problem statement: A steel ball bearing of mass 3.3 × 10^–5 kg and radius 1.0 mm displaces 4.1 × 10^–5 N of water when fully immersed. The ball is allowed to fall through the water until it reaches its terminal velocity. Calculate the terminal velocity if the viscosity of the water is 1.1 × 10^–3 N s m^–2.

Solution:

F = W – U

6πη r ν = W – U

v = (W – U)/(6πηr)

v =(3.3 × 10^–5 kg × 9.8 N kg^–1 – 4.1 × 10^–5 N ) / (6π × 1.1 × 10^–3 N s m^–2 × 1.0 × 10^–3 m)

= 14 m/s

Answer: The terminal velocity is 14 m/s