# Derive Third Equation of Motion.

By Ritesh|Updated : November 9th, 2022

The third equation of motion is v2 - u2 = 2as. Physics defines equations of motion as ones that express how a physical system behaves based on how it moves through time. It is possible to determine components like velocity (initial and final), displacement(s), time(t), and acceleration using one of three equations of motion (a). The three equations of motion are as follows:

• The first equation of motion is given by v = u + at
• The second equation of motion is given by s = ut + ½ at2
• The third equation of motion is given by v2 = u2 + 2as

### Equations of Motion

The following methods can be used to obtain the equations of motion:

• By Graphical Method
• By Calculus Method
• By Simple Algebraic Method

### Third Equation of Motion

We need to demonstrate the third motion equation.

The third Equation of Motion states that:

v2 - u2 = 2as

Proof -

• Let initial speed = u
• Acceleration = a
• Final speed = v
• Time = t

From the first equation of motion we have

v = u + at

On rearranging we get

or t = (v - u)/a

Average velocity is given as (v + u)/2

Also, we know that

Distance = average velocity X Time

s = (v + u)/2 x (v - u)/a

After solving

We get

v2 - u2 = 2as

This is the third equation of motion.

Summary:

## Derive Third Equation of Motion.

The third equation of motion is v2 - u2 = 2as. The equation of motion can be determined using calculus, graphical and simple algebraic methods.