Terminal Velocity Formula: Derivation & Examples

The terminal velocity is the highest constant velocity attained by a body when falling through a viscous medium. When the force of resistance of a medium equals and opposes the force of gravity, it is said to be accomplished. 

• Terminal Velocity is liable to drag force, which increases with an increase in velocity.
• In this sum, the total of the buoyancy and drag force is equal to the force of gravity moving downward, affecting the object.
• The acceleration of the object is considered zero since the net force affecting the object is zero. 
• Driving force is used to make objects fall under the effect of gravity.
• Terminal velocity is also known as settling velocity.
• In this, the force of air is directly proportional to the velocity of the object falling.
• The resistance of air is equal to the magnitude of the weight of the object falling.
• Raindrops and mists of tiny droplets are the most common examples of terminal velocity in real life.
• The formula of the velocity is as follows:

V_T = √2mg / AC_d

Key Terms: Terminal Velocity, Force, Velocity, Speed, Weight, Resistance,  Gravitational Force, Gravity, Drag Force, Fluid Density, Viscosity, Fluid Dynamics, Mass

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What is Terminal Velocity?

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Terminal velocity is a type of velocity that is attainable when an object falls through a fluid. An object is considered to be moving at terminal velocity in fluid dynamics if its speed remains constant.

• A restraining force is put on it by the fluid in which it moves.
• The qualities of the fluid and the mass of the object affect the object's terminal velocity.
• It is also affected by the projected cross-sectional surface area of the object.
• The net force acting on the object is zero.
• The use of parachutists can explain the concept of the terminal velocity.
• When the parachutist opening is delayed by 150 miles per hour.

Terminal Velocity Explanation

When a thing falls from a height, we find:

• At first, it falls at a fast speed, as the force of gravity causes it to speed up rapidly.
• Its body weight acts downward, air resistance pushes upward, and gravitational force pulls downward.
• After a while, it falls at a certain constant speed. 
• This happens as the air drag force is exactly equal to the gravitational force.
• The object will no longer speed up or decelerate if these two forces are perfectly balanced.
• However, it will continue to descend at a steady pace.
• This steady speed is terminal velocity.

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How to find Terminal Velocity?

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Any moving object can attain terminal velocity if its speed is constant against the force exerted by the fluid through which it is moving. It is calculated by the formula:

V_T = √2mg / AC_d

• V_T stands for terminal velocity;
• M denotes the mass of the falling body;
• G acceleration because of gravity;
• A denotes the projected area of the item;
• Ρ is the fluid density;
• C_d denotes the drag coefficient.

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Drag Force

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Drag force is a resistive force that works in the opposite direction of an object's relative motion in relation to a fluid. It depends on the object's velocity as it falls through the fluid, which makes it different from other resistive forces.

• It is termed as air resistance or fluid resistance.
• Drag D denotes the drag coefficient.
• The complicated dependencies can be described in a single variable, called the drag coefficient, represented as Cd.
• Force is Cd multiplied by density r multiplied by half the velocity V squared multiplied by the reference area A.
• The drag coefficient's value can be determined by experiments.

Drag Force Equation

The equation for drag force is given as:

FD = ½ \(\rho\)\(\nu\)²C_DA

• FD = Drag
• \(\rho\)= fluid density
• \(\nu\) = speed of the object in relation to the fluid
• C_D = Drag coefficient
• A = cross sectional area

Factors affecting Drag Force

The factors affecting drag force are as follows:

• Density
• Compressibility
• Viscosity of the fluid
• Square of the object's velocity
• Shape and size of the body
• Leaning of the body towards the flow

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Terminal Velocity Derivation

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Terminal Velocity Derivation gives:

D = 1/2ρv²AC_d

• Net force exerted on the body-
• F_net = ma = Gravitational force - Drag force
• ma = mg - 1/2ρv2AC_d
• At equilibrium, F = 0, this implies:
• 0 = mg - 1/2ρv²AC_d
• mg = 1/2ρv²AC_d
• By solving this,
• V_T = √2mg / ρAC_d

It can be derived as  W= weight of the ball,

• W= mg
• W= vρ_og (v= volume)
• F_V = Viscous force
• F_V=6πηrv
• F_d = buoyant force,
• F_d = vρg
• In Equilibrium,
• W= F_V + F_d
• F_V = W - F_d
• 6πηrv = vρ_og – vρg (v= 4/3πr³) 
• 6πηrv = 4/3πr³ (ρ_o−ρ) g

v= 2r² (ρ_o−ρ) g / 9η

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Things to Remember

• Terminal Velocity is the highest constant velocity attained by a body while falling during a thicker medium.
• It is also equal to the maximum velocity of a body travelling through a viscous fluid.
• It is achieved when the medium's force of resistance equals and opposes the force of gravity.
• As the velocity rises, the retarding force rises with it will be reached when gravity's force equals the resistance force.
• Formula of terminal velocity is given as: V_T = √2mg / AC_d

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