Terminal Velocity Derivation

The maximum speed or velocity that an object can attain as it falls through the fluid is called terminal velocity. It happens when the sum of drag force from the fluid and buoyancy is equal to downward gravitational force acting on the object. In these situations, the net force that acts on the object is 0. This is the reason why the object has 0 acceleration. 

In fluid dynamics, the object is moving at terminal velocity if the object speed is constant. The reason behind speed being constant is the restraining force which is exerted on an object by fluid via which that specific object is moving. It is possible to change the terminal velocity of an object by the fluid properties, object’s predicted cross-sectional surface area, and mass of the object. Let us have a look at the Mathematical Representation of the terminal velocity and derivation of the terminal velocity.

The Mathematical Representation of the Terminal Velocity

Using Mathematical terms, the formula for derivation of terminal velocity without the buoyancy effects can be described as:

Vt = \[\sqrt{\frac{2mg}{pACd}}\]

Here, Vt depicts terminal velocity, ‘mis’ represents mass of falling object, ‘g’ is the acceleration of the object because of velocity, ‘Cd’ is the drag coefficient, ‘p’ stands for fluid density via which the object is falling, and ‘A’ is the protection of object. In reality it is considered that the object can attain terminal velocity asymptomatically. The effect of buoyancy is upward force which is acting on the object from the surrounding field. The effect of buoyancy can be considered within the formula using Archimedes principle.In the equation, the mass m needs to be reduced by displaced fluid’s mass, that is pV, where V is the object volume. After that, take Mr = m - pV instead of m in all of the subsequent formulas. 

Derivation of Terminal Velocity

The solution for answering the question ‘How to derive the expression for terminal velocity?’ is provided in the Vedantu notes on “Terminal Velocity Derivation”. The derivation of terminal velocity using Mathematical terms according to the drag equation comprehensive stepwise equations to arrive at the desired equation. First we use the formula for the free fall of the object then we derive the equation assuming that free fall happens in the positive direction. Next we derive the differential form of the equation, and then we integrate the equations. The complete stepwise derivation is available with Vedantu notes that covers the different aspects related to “Terminal Velocity Derivation”.

Terminal Velocity in Presence of Buoyancy Force

When the effects of buoyancy are considered, then the object that is falling through the fluid beneath its own weight can possibly reach the terminal velocity, if net force that is acting on the specific object becomes 0. When this object has obtained terminal velocity, the object weight gets balanced by the force of upward buoyancy, and the drag force. This is represented by,

1. W = Fb + D

Here, W is object weight, Fb is buoyancy force, and D is drag force which acts on the object. 

If the falling object has spherical shape, then expressions for the forces (W, Fb, D) are as follows.

2. W = \[\left ( \frac{\pi }{6} \right )d^{3}\] PsG,

3. Fb = \[\left ( \frac{\pi }{6} \right )d^{3}\] PG

4. D = Cd(½) pV \[^{2}\] A

How is Terminal Velocity Useful?

When an object is dropped from rest, its speed will increase until it reaches terminal velocity. When an object is forced to move faster then its terminal velocity will slow down and reach to its constant velocity upon release.Terminal velocity occurs when the speed of a moving object is no longer increasing or decreasing and the object’s acceleration or deceleration is zero.

Objects Fall at Different Speeds- Why?

When the object falls down, the speed of the falling object increases continuously. As the speed of the falling object gets faster and faster, the air drag force increases and it is equal to the force of gravity. Thus, there is no net force acting on the object. When two forces acting on an object are balanced, the object will neither speed up or slow down but it will continue falling at a constant velocity and this is called the terminal velocity. The air drag force depends on the size and shape of the object. Objects having a large surface area such as in a parachute will have a much lower terminal velocity than objects with a smaller surface area such as a person falling from a plane. The weight of an object also affects the air drag force on an object and its terminal velocity. This also explains why a crumpled piece of paper will fall faster than a flat piece of paper. The weight of the paper is the same but the air drag force reduces because the surface area and drag coefficient change. This produces higher terminal velocity of the crumpled paper than the flat paper.