# 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 ‘v’ 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.