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

By BYJU'S Exam Prep

Updated on: September 25th, 2023

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


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


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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