Terminal Velocity Derivation: Definition & Applications

Terminal velocity, a fundamental concept in fluid dynamics, is the maximum speed attained by an item when moving through media such as air or liquid. 

• This velocity represents the moment at which an object's acceleration reaches zero, hence stabilising its speed. 
• Surface area, viscosity, density, and mass are all elements that influence terminal velocity.
• Skydivers use this idea to guide their descent, aligning their bodies in order to achieve a steady fall.

Key terms: Terminal Velocity, Highest Speed, Velocity,, Buoyancy, Constant Speed, Fluid, Cross-Sectional Surface Area

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Terminal Velocity: Definition

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Terminal velocity refers to the highest speed an object can reach when moving through a medium, such as air or liquid. It is attained when a moving object's speed is no longer increasing or decreasing. 

• Hence, the speed of its acceleration becomes zero. 
• Since the force of air resistance is approximately equivalent to the falling object's speed, air resistance increases for an object that accelerates after being dropped from rest 
• At terminal velocity, air resistance equals the weight of the falling object in magnitude. 
• The two forces are oppositely oriented, the overall force on an object is zero, and the object's speed has become constant. 
• Terminal velocity depends on an object's cross sectional surface area, coefficient of viscosity of the medium, object density, density of the medium and mass of the object. 

Read More: Bernoulli’s Principle

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Applications of Terminal Velocity

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Applications are mentioned below:

• Raindrops have a considerably lower terminal velocity, while a mist of tiny oil droplets has an even lower terminal velocity. 
• An item dropped from rest will accelerate until it achieves terminal velocity
• An object pushed to go faster than terminal velocity will slow down to this constant velocity upon release.
• Skydivers use terminal velocity during the free fall from the aircraft. 
• The terminal velocity is the downward speed achieved during freefall despite the resistance of the air. 
• Skydivers properly gear themselves and practice to modify their body posture so that they can go at the same pace. 
• When they depart an airplane, their bodies accelerate until air resistance equals gravity, resulting in a steady rate of descent.

Read More: Properties of Fluids

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Terminal Velocity: Derivation

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Suppose the density of an object as ρo, which attains terminal velocity Vt, while falling through the liquid of density ρ. The net force exerted on the item at terminal velocity is zero. 

The drag force and buoyant force cancel out the gravitational force, leading to no acceleration.

Terminal Velocity 

Where, 

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

hence, the terminal velocity acquired by the ball of radius r, when dropped through a liquid of viscosity η and density ρ is V_t = 2r² (ρ_o−ρ) g / 9η

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Solved Examples

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Example 1: Calculate the terminal velocity of a raindrop with a radius of 0.2 mm falling through air. The density of air is 1.2 kg/m³, and the viscosity of air is 1.8×10−5N s/m².

Ans: Using the terminal velocity formula:

V_t = 2r² (ρ_o−ρ) g / 9η

Substitute the given values:

V_t = 2 × (0.20*10^-3 )² × (1.2-1.2) × 9.81 / 9 × 1.8×10⁻⁵

V_t = 0 m/s

Example 2: A small plastic ball with a radius of 2 cm and a density of 900 kg/m³ is dropped from rest in a pool of water. The viscosity of water is 1.0×10−3N s/m². Calculate the terminal velocity of the ball.

Ans: Using the terminal velocity formula:

V_t = 2r² (ρ_o−ρ) g / 9η

Substitute the given values:

V_t = 2 × (0.02)² × (900-1000) × 9.81 / 9*1.0×10⁻³

V_t = 0.087 m/s

Read More:

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Things to Remember

• Terminal velocity is the maximum constant velocity obtained by a body while falling through a thick material.
• The maximum velocity of a body travelling through a viscous fluid is called terminal velocity.
• The formula of terminal velocity is, V_t = 2r² (ρ_o−ρ) g / 9η
• Objects dropped from rest accelerate until they reach terminal velocity due to gravity and drag.
• Objects pushed beyond terminal velocity slow down to the terminal velocity when released, as drag counters their speed.
• At terminal velocity, air resistance equals the weight of the falling object in magnitude. 
• Raindrops have a considerably lower terminal velocity, while a mist of tiny oil droplets has an even lower terminal velocity. 

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