# Derive the Third Equation of Motion: v2 - u2 = 2as

By K Balaji|Updated : November 9th, 2022

The phenomenon of motion is when an item shifts positions. Displacement, distance, velocity, acceleration, speed, and time are all used to describe motion.

### First Equation of Motion

The first equation of motion connects acceleration, time, and velocity.

v = u + at

This is the first equation of motion where,

a = acceleration

u = initial velocity

v = final velocity

t = time taken

### Second Equation of Motion

The second equation of motion connects time, displacement, velocity, and acceleration. The body's displacement is depicted by the region underneath the v-t graph.

s = ut + ½ at2

Step 1: To prove:

v2 - u2 = 2as

Where

t is the time.

u is initial velocity,

v is final velocity,

a is the acceleration

and s is the displacement,

Step 2: Derivation:

We employ both of the equations of motion to obtain the third equation of motion.

s = ut + ½ at2 ….. (1) [Second equation of motion]

v = u + at …. (2) [First equation of motion]

On rearranging we get

t = (v - u)/a

Substitute the value of t in equation (1) s = ut + ½ at2

We get,

s = u (v - u)/a + ½ a (v-u/a)2

2as = 2u (v - u) + (v - u)2

2as = 2uv - 2u2 + v2 - 2uv + u2

On rearranging we get

2as = v2 - u2

v2 - u2 = 2as

Summary:-

## Derive the Third Equation of Motion: v2 - u2 = 2as

The third equation of motion is v2 - u2 = 2as. It is an equation which connects the final velocity, initial velocity, acceleration and distance.

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