How to get these SUVAT formulae? When learning kinematics, I have learned these three formulae
$s=1/2(u+v)t$
$s=ut+1/2at^2$
$v^2=u^2+2as$


*

*s = displacement

*u = initial velocity

*v = final velocity 

*t = time

*a = acceleration


My teacher then say that we can get the second and third formulae from the first formulae and the formulae of acceleration
Proof for $s=ut+1/2at^2$
From the formulae $a=(v-u)/t$ and arrange you get $v=u+at$
Then substitute $v=u+at$ into $s=1/2(u+v)t$ 
$s=1/2(u+v)t$
$s=1/2(u+u+at)t$
$s=1/2(2u+at)t$
$s=1/2(2ut+at^2)$
$s=ut+1/2at^2$
Proof for $v^2=u^2+2as$
From the formulae $a=(v-u)/t$ and arrange you get $t=(v-u)/a$
Then substitute $t=(v-u)/a$ into $s=1/2(u+v)t$ 
$s=1/2(u+v)t$ 
$s=1/2(u+v)*(v-u)/a$ 
$s=(v^2-u^2)/2a$
$2as=v^2-u^2$
$v^2=u^2+2as$
So, my questions is if we can get the second and third formulae from the first formulae, how do we get the first formulae?
 A: The first one is basically derived from the velocity-time graph. As the acceleration is constant the area under the graph represents displacement (s). 
Thus by the formula to get the area under a trapezium.
$$\text{Area} =\frac{1}{2} ( \text{sum of the parallel sides}) \times ( \text{vertical height})$$.
We have 
$$S = \frac{1}{2} ( v + u)\cdot t$$.
A: Suppose you have a constant acceleration $a$. Acceleration is the derivative of velocity with respect to time, so we have:
$$ \frac{dv}{dt} = a $$
and if we integrate this we get:
$$ v = at + C $$
where $C$ is the constant of integration. To find $C$ we note that when $t = 0$ the velocity is equal to $u$ (the initial velocity) so that means $C = u$ and the full equation is:
$$ v = u + at $$
Now, velocity is the derivative of distance with time, so we have:
$$ \frac{ds}{dt} = u + at $$
Again, we can integrate this to get:
$$ s = ut + \tfrac{1}{2}at^2 + C $$
where again $C$ is a constant of integration. This time we use the condition that at $t = 0$ the distance $s$ is zero, and this means $C = 0$. So we have derived your second equation:
$$ s = ut + \tfrac{1}{2}at^2 $$
We then get the first and third equations from the second one.
