# Equation of Motion

- First equation of motion
- Second equation of motion
- Third equation of motion

Relation among velocity, distance, time and acceleration is called equations of motion. There are three equations of motion, viz. first, second and third equations of motion. In this part, you will learn about mathematical derivation of equations of motion.

## First Equation of Motion

The final velocity (v) of a moving object with uniform acceleration (a) after time, t.

Let, the initial velocity = u.

Final velocity = v.

Time = t

Acceleration = a

We know that, Acceleration (a) `=(text{Change in velocity})/(text{Time taken})`

`=> a=(text{Final velocity-Initial velocity})/text{Time taken}`

`=>a=(v-u)/t`

`=>at=v-u`

`=>at-v=-u`

`=>-v=-u-at`

`=>v=u+at` ---(i)

This equation is known as first equation of motion.

## Second Equation of Motion

Distance covered in time (t) by a moving body.

Let, Initial velocity of the object = u

Final velocity of the object = v

Acceleration = a

Time = t

Distance covered in given time = s

We know that,

Average velocity `=(text{Initial velocity+Final velocity})/2`

∴ Average velocity `=(u+v)/2` ----(ii)

We know that, Distance covered (s) in given time = Average velocity x Time

Or, s = Average velocity x Time -----------------(iii)

After substituting the value of average velocity from equation (ii) we get

`=>s=(u+v)/2xxt`

After substituting the value of āvā from first equation of motion we get,

`=>s=(u+(u+at))/2xxt`

`=>s=(u+u+at)/2 xxt`

`=>s=(2u+at)/2 xxt`

`=> s=(2ut+at^2)/2`

`=>s=(2ut)/2+(at^2)/2`

`=>s= ut+(at^2)/2`

`=>s=ut+1/2 at^2` ----(iv)

The above equation is known as Second equation of motion.

## Third Equation of Motion

The third equation of motion is derived by substituting the value of time (t) from first equation of motion.

We know from first equation of motion, `v=u+at`

`=>v-u=at`

`=>at=v-u`

`=>t=(v-u)/a` -----(v)

We know that the second equation of motion is, `s=ut+1/2at^2`

By substituting the value of `t` from euqation (v) we get

`s=u((v-u)/a)+1/2a((v-u)/a)^2`

`=>s=uxx(v-u)/a+1/2a((v-u)^2)/a^2`

`=>s=(u(v-u))/a +(axx(v-u)^2)/(2xxaxxa)`

`=>s=(uv-u^2)/a + ((v-u)^2)/(2a)`

`=>s=(2(uv-u^2)+(v-u)^2)/(2a)`

`=>2as=2uv-2u^2+v^2+u^2-2uv`

`=>2as=-2u^2+v^2+u^2`

`=>2as=-u^2+v^2`

`=>2as+u^2=v^2`

`=>v^2=u^2+2as` ---(vi)

This is called the Third equation of motion.