How to derive the equations of Motion

We now wish to derive the equations of Motion for a body travelling along a straight line with uniform acceleration.

First Equation of Motion :

we can remember that acceleration =change in velocity /time
Now using,

u- initial  velocity
v- final velocity
t- time taken
a- acceleration
Since change in velocity means final velocity - initial velocity, we now have
acceleration =final velocity - initial velocity /time
substituting
a=v-u/t or v=u+at

Second Equation of Motion. 

Let (s) be the distance covered in a time (t) with the initial and final velocities given by u and v respectively.
Average velocity = u+v/2.
but we now know that v=u+at, hence
average velocity = u+(u+at)/2=(2u+at)/2 = u +at/2.
Distance covered (s) =average velocity ×time
s= (u+at/2)t = ut+at²/2.

Third Equation of Motion. 

We can obtain the third equation of Motion with use of the first and second equation of Motion. From the first equation we have: v=u+at.
From the second we have
s=ut+at²/2.
if we square both sides of the first equation, we have
v²=(u+at)²=u²+2uat+a²t²
v²= u²+2a[ut+at²/2]
but s= ut+at²/2,hence we have
v²=u²+2as.
Also s= 1/2(v+u)t

The three equations of Motion with uniform acceleration are:
v=u+at
s=ut+at²/2 or 1/2(v+u)t
v²=u²+2as
where
u=initial velocity
v=final velocity
t=time
s=distance
a=uniform acceleration.
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