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Terminal Velocity Formula

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Introduction

The terminal velocity formula is an important derivative of fluid mechanics. When an object is immersed in the fluid, many forces will be acting on it simultaneously, such as the gravitational force which is acting in the downward direction, normal force, and the viscous force acting in an upward direction. Further, the action of all these forces will result in the drag force. 


The air drag force generated will be exactly equal to the force of gravity acting on the object in a downward direction. If these two forces i.e., the gravitational force and the air drag force are exactly balanced (i.e., apparently equal), then the object will no longer get accelerated or decelerated, but the object will continue falling with some constant velocity. The constant velocity with which the object is falling into the viscous fluid or viscous medium is called the terminal velocity. In this article, we will look into the terminal velocity formula or terminal speed formula, the formula of terminal velocity derivation along with the solved numerical problems.

Formula Of Terminal Velocity

So now, what is terminal velocity? 


The terminal velocity is the constant velocity achieved by the object when it attains the balanced forces. Since the air drag force completely relies on the shape and size of the object, i.e., radius and mass of the object. Thus objects with a huge surface area are found to have a considerably lower terminal velocity than objects with a less surface area. At the same time along with the size and the shape of the object, even the weight of the object will affect the air drag force on the object and, thus, it will even affect the terminal velocity of the object.


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As the speed of the object under motion increases, at the same time, the drag force acting on it will also increase. The increase or decrease in the air drag force also depends on the substance, whether it is passing through mediums such as air or water. At a certain speed, this air drag force of resistance or the air resistance will be equal to the gravitational pull on the object. At this condition, the object will reach the constant terminal velocity.

Terminal Speed Formula

Therefore, the terminal velocity of an object is defined as the maximum velocity attained by an object that is falling through a fluid medium and it can be calculated using the terminal velocity formula or terminal speed formula. It is witnessed when the total sum of drag force and buoyancy is comparable and almost equal to the downward gravitational force that is acting on the object. We have to note that, the acceleration of the object is null or zero as the net force acting on the object is zero, due to constant velocity.


In fluid mechanics, for an object to accomplish its terminal velocity should have a constant velocity against the net force exerted by the fluid medium through which it is moving. This term terminal velocity though it is a physical parameter yet it is made popular by skydivers.


We know that, according to Stokes law of fluid mechanics, it states that the viscous force acting on a small sphere falling through a viscous medium or the fluid is directly proportional to the radius of the sphere, velocity, and coefficient of viscosity. The constant velocity with which the object is falling into the viscous fluid or viscous medium is called the terminal velocity. The terminal velocity formula in fluid mechanics is given by:


\[\Rightarrow v_{terminal} = \frac{2\ r^{2}(\rho - \sigma)g}{9\eta}\]


Where,


\[\rho, \sigma\] - The densities of the fluid and the sphere ball


r - The radius of the sphere


\[\eta\] - The coefficient of viscosity


g - The acceleration due to the gravity


Now, let us have a look at the formula of terminal velocity. The terminal velocity of an object mainly depends on the gravitational pull and the height from which it is dropped. The terminal velocity formula is given by:


\[\Rightarrow v_{terminal} = \sqrt{2gh}\ m/s\]


Where,


g - The acceleration due to gravity


h - The height from the ground


This is the formula of terminal velocity or the terminal speed formula in fluid mechanics. It can be used to estimate the terminal velocity of any object passing through a viscous fluid medium.

Examples:

  1. Calculate the height of the body which is dropped with a terminal velocity of 120 m/s.

Sol:

Given,


The terminal velocity of the object= \[ v_{terminal} = 120\ m/s\]


The acceleration due to gravity=  \[g = 9.8\ m/s^{2}\]


Now we are asked to determine the height from which the object is being dropped. We know that the formula of terminal velocity is given by:


\[\Rightarrow v_{terminal} = \sqrt{2gh}\ m/s\]……..(1)


Where,


g-The acceleration due to gravity


h-The height from the ground


Squaring on both sides of the equation (1) and rearrange the equation for the height of the object we get:


\[\Rightarrow v^{2}_{ter} = 2\ g\ h\]


\[\Rightarrow h = \frac{ v_{ter}^{2}}{2\ g}\]……(2)


Substituting all the required data in equation (2) and simplify:


\[\Rightarrow h = \frac{ v_{ter}^{2}}{2\ g} = \frac{(120)^{2}}{2 \times 9.8}\]


\[\Rightarrow h = 734.69 m \simeq 735m\]


Therefore, the object was dropped from a height of 735 mabove ground level.

  1. Consider two spheres A and B each of density 8 g/cm3 and the diameters of 1cm and 0.5cm.Sphere A is dropped into a liquid of density 0.8 g/cm3 and viscosity \[\eta\] = 3poiseulles. Sphere B is dropped into a liquid of density 1.6 g/cm3 and viscosity \[\eta\] = 2 poiseulles. The ratio of the terminal velocities of A and B is: (IIT JEE 2016)

Sol:

Given,


The densities of sphere A and B= \[\rho_{A}\ and\ \rho_{B} = 8g/cm^{3}\]


The density of the liquid in which sphere A is dropped= \[\sigma_{A} = 0.8g/cm^{3}\]


The viscosity of the liquid in which sphere A is dropped= \[\eta_{A} = 3\]


The density of the liquid in which sphere B is dropped= \[\sigma_{B} = 0.8g/cm^{3}\]


The viscosity of the liquid in which sphere B is dropped= \[\eta_{B} = 2\]


The radius of the sphere A= \[r_{A} = 1\ cm\]


The radius of the sphere A \[r_{B} = 1\ cm\]


Therefore, the terminal velocity of sphere A is given by:


\[\Rightarrow v_{A} = \frac{2\ r^{2}(\rho - \sigma)g}{9\eta}\]……(1)


Where,


\[\rho, \sigma\] - The densities of the fluid and the sphere ball


r - The radius of the sphere


\[\eta\] - The coefficient of viscosity


g - The acceleration due to the gravity


Substituting the values in equation (1), and simplify:


\[\Rightarrow v_{A} = \frac{2\ r^{2}(\rho - \sigma)g}{9\eta} = \frac{2(1)^{2} (8-0.8)(9.8)}{9(3)}\]


\[\Rightarrow v_{A} = 5.226\ m/s = 5\ m/s\]


Similarly, the terminal velocity of the sphere B is given by:


\[\Rightarrow v_{B} = \frac{2\ r^{2}(\rho - \sigma)g}{9\eta} = \frac{2(0.5)^{2} (8-1.6)(9.8)}{9(2)}\]


\[\Rightarrow v_{B} = \frac{2(0.5)^{2} (8-1.6)(9.8)}{9(2)}\]


\[\Rightarrow v_{B} = 1.75\ m/s\]


Now, we are asked to determine the ratio of terminal velocities of sphere A and B:

\[\Rightarrow \frac{v_{A}}{v_{B}} = \frac{5}{1.75}\]


\[\Rightarrow \frac{v_{A}}{v_{B}} = 2.85 \simeq 3\]


Therefore, the ratio of terminal velocities of the sphere A and B is 3.


FAQs on Terminal Velocity Formula

1. What is the terminal velocity in physics?

The terminal velocity in physics is the maximum velocity attained by the object when air drag force and the gravitational pull are balanced and equal to each other.

2. What is the terminal velocity formula in fluid mechanics?

The terminal velocity formula in fluid mechanics is given by:

\[\Rightarrow v_{terminal} = \frac{2\ r^{2}(\rho - \sigma)g}{9\eta}\]


Where,


\[\rho, \sigma\] - The densities of the fluid and the sphere ball


r - The radius of the sphere


\[\eta\] - The coefficient of viscosity


g - The acceleration due to the gravity