# Motion

## Equation of Motion:

Relation among velocity, distance, time and acceleration is called equations of motion. There are three 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.