How can we derive three equation …

First Equation:
Acceleration is defined as the rate of change of velocity.
Let, V = final velocity; V_o = initial velocity, T= time, a =acceleration.
By definition of acceleration:
{tex}a = \frac{{V - {V_o}}}{T}{/tex}
{tex}at = V - {V_o}{/tex}
{tex}V = {V_o} + {\rm{ }}at{/tex}
Since, V_o =u = initial velocity
therefore V = u +at
Second Equation:

Let at time T=0, body moves with initial velocity u and at time ‘t’ body has final velocity ‘v’ and in time ‘t’ it covers a distance 'S'.
AC = v, AB = u, OA = t, DB = OA = t, BC = AC - AB = v - u
Area under a v-t curve gives displacement so,
S= Area of triangle DBC + Area of rectangle OABD .......(i)
Area of {tex}\Delta DBC = \frac{1}{2} \times Base \times Height{/tex}
= {tex}\frac{1}{2} \times DB \times BC{/tex}
= {tex}\frac{1}{2} \times t \times \left( {v - u} \right){/tex} ......... (ii)
Area of rectangle OABD {tex}= length \times breadth{/tex}
= {tex}OA \times BA{/tex}
= {tex}t \times u{/tex} ......... (iii)
From (i), (ii) and (iii)
S= ut+ {tex}\frac{1}{2} \times t \times \left( {v - u} \right){/tex}
S= ut + {tex}\frac{1}{2} \times t \times at{/tex}
S= ut + {tex}\frac{1}{2}a{t^2}{/tex}​​​​​​​
Third Equation:
​​​​​​​Let at time t=0, the body moves with an initial velocity u and time at ‘t’ has final velocity ‘v’ and in time ‘t’ covers a distance ‘s’
Area under v-t graph gives the displacement

S = Area of {tex}\Delta{/tex}DBC + Area of rectangle OABD
S = {tex}\frac{1}{2} \times base \times height + length \times breadth{/tex}
{tex}S = \frac{1}{2} \times DB \times BC + OA \times AB{/tex}
{tex}S = \frac{1}{2} \times t \times (v - u) + t \times u{/tex} ........(i)
Now, v - u = at
 {tex}\frac{{v - u}}{a} = t{/tex}
put the value of 't' in equation (i)
{tex}S = \frac{1}{2} \times (v - u)\frac{{(v - u)}}{a} + u \times \left( {\frac{{(v - u)}}{a}} \right){/tex}
{tex}S = \frac{{{{(v - u)}^2}2u(v - u)}}{{2a}}{/tex}
{tex}S = \frac{{{v^2} + {u^2} - 2uv + 2uv - 2{u^2}}}{{2a}}{/tex}
{tex}S = \frac{{{v^2} - {u^2}}}{{2a}}{/tex}
{tex}2as = {v^2} - {u^2}{/tex}