Here we will discuss in details about **terminal velocity **and **Terminal Velocity Equation**. Before that we will also cover a few important pointers on **free fall** and then discuss on Air Drag, Drag force and Drag Force Equation. In one of earlier posts we have discussed about the free fall equations. You can check that if you want at this point.

## Terminal Velocity – understanding of other deciding factors

### What is a free fall?

From the name itself, we can see, there are 2 things. ‘Fall’ and ‘Free fall’.

We know objects, when thrown normally upwards, comes downwards after a while, so it’s a fall.

And we all know the reason behind this fall. That is Gravity or Gravitational force between the earth and the object in discussion.

**Now ‘Free fall’ means, the fall of the object is free from all resistance.**

**In Newtonian motion, Free fall is the motion of a body when the sole force acting on the body is the Gravity. ****So when we discuss a free fall or solve a numerical related to it, we consider air resistance as zero.**

### Free fall equations

The free fall equations resemble with the equations of linear motion, here in place of acceleration (a) we use the **acceleration due to gravity** (g). You may go through this post to read in details:**Equations**

**Free Fall and Weightlessness:**

During Free fall as there is no surface to provide the Normal Reaction Force on the freely falling body, the falling body has the feeling of weightlessness. But remember that, this is a feeling only and gravity continues to work on the falling body. Read on weightlessness.

### Practicality of a Free Fall

Now practically we can consider air resistance as zero only when it’s found Negligible! Couple of situations can be considered for this.

1) height from where the fall is happening is very small.

2) speed of the falling object is small.

3) area of the surface facing air resistance is small.

4) density of the falling object is considerably high.

5) shape of the object is such (aerodynamic) that it cuts through air without much resistance.

When an object is dropped from a height and that in vacuum then this free fall is observed actually!

### Drag Force

Now when an object is falling through the atmosphere from a high altitude, then we can no more ignore the air resistance force.

This means 2 forces are now acting simultaneously on the object.

As per Newtonian concepts, it’s no more a Free fall.

An opposing resistive air force, also called as Drag Force, is now active on the falling object, apart from the downward gravity force.

*Force 1:***Gravity** which is equal to the weight (W) of the falling body. If the mass of the body is m and g is the acceleration due to gravity (9.8 m/s^2), then Gravity is equal to the weight W of the object, where

Gravity=W= mg…… (1)

** Force 2:** The

**Air Drag**is like frictional force opposing the motion but its formula is different from Friction. This air drag depends on few factors, which are all present in its equation.

**Drag Force equation**

**Drag Force = D = 0.5 * K * r * V ^{2} * A … (2) **

**[**This is the

**Drag force formula ]**

here K = Drag Coefficient of the falling object (it depends on the inclination of the shape and some other criteria like air flow)

r = air density

V= velocity of the falling object

A = cross sectional area of the object falling

**Here we can see that this Drag increases with the increase in velocity (as square of the velocity).**

### Gravity Vs Drag

Now for a falling body, Gravity and this Drag force works in opposite direction.

**Hence the net force on the falling object is F = W – D ….. (3)**

As there is a net force on the falling body, it will certainly have an acceleration, which is **a = (W – D) / m**

**Please note that this acceleration is different from g (acceleration due to gravity).**

As the body is under acceleration, so certainly its velocity will rise with time. As the velocity increases, the Drag Force goes on increasing. (See equation 2)

W remains constant throughout the fall.

At some point, if time of the fall permits, **D becomes equal to the weight or gravity W**.

Here the net force on the falling object becomes zero and acceleration becomes 0.

That means now onwards the falling object falls with a constant velocity.

W – D = 0;

**so, W = D ……………………………. (4)**

## Terminal Velocity

Net force on the falling object if becomes zero means, no acceleration of the falling object at that moment.

As per Newton’s First Law, in this scenario the object falls with a constant downward velocity.

This constant velocity directing vertically downwards is called** Terminal Velocity**.

So what we have seen here is that

**A falling object, under the influence of gravity and Drag force, falls with an acceleration, which is not equal to g.**

**As the falling object gains velocity while falling, it attains a velocity called Terminal Velocity.**

**Once it attains the terminal velocity, Net Force on it becomes zero. As a result its downward velocity becomes constant i.e. at terminal velocity acceleration is zero.**

### Terminal Velocity Equation

From the above equations, we get the **equation for Terminal Velocity**.

In this case, W = D

so, W = 0.5 * K * r * **V**^2 * A

Here we get **Terminal Velocity Equation or formula:**

**Terminal Velocity = ****V =[(2 * W) / (K*r*A)] ^{1/2} **

…..

**…(5)**

**[formula for Terminal Velocity]**

here K = Drag Coefficient of the falling object (it depends on the inclination of the shape and some other criteria like air flow)

r = air density

W = weight of the falling object

A = cross sectional area of the object falling

### Terminal Velocity – Factors

**Terminal Velocity varies directly with the weight of the object. (see equation 5)**

Let’s take 2 objects with same shape and size but with different weight.The heavier object will attain higher terminal velocity.That means the heavier object falls faster than its lighter contestant, once terminal velocity is attained by both the objects.(**please note that, in vacuum where there is no air resistance or drag, both these objects would fall with same velocity)

**An object with a large cross-sectional area will fall slower than an object with smaller area. (see equation 5, Area A is in denominator)**

**Again an object with higher drag coefficient falls slower than an object with lower drag coefficient. (see equation 5, K is in denominator)**