Derivation of the kinematic equations

We start with the definitions of average acceleration and average velocity,

$\bar a=\dfrac{\Delta v}{\Delta t}$

$\bar v=\dfrac{\Delta x}{\Delta t}$.

The kinematic equations are derived under the assumption that the acceleration is constant. When the acceleration is constant, the average and instantaneous accelerations are the same, so we can replace $\bar a$ with $a$.

We now make a few changes to the notation. First, we take the initial time $t_1$ to be 0 and the final time $t_2$ to be $t$, so the time interval is $\Delta t=t_2-t_1=t-0=t$.

Next, we take the initial velocity to be $v_1=v_0$ and the final velocity to be $v_2=v$. So $v$ is now the velocity after a time interval of $t$, that is, the velocity at time $t$. The change in velocity is therefore $\Delta v=v_2-v_1=v-v_0$.

Putting these into the average acceleration equation:

$a=\dfrac{v-v_0}{t}$.

Solving for $v$,

$\boxed{v=v_0+at}$

This is the first kinematic equation.

Next, take the average velocity equation and substitute $\Delta t =t$ to get

$\bar v=\dfrac{\Delta x}{t}$.

From this,

$\Delta x=\bar v t$.

We have two velocities, the initial velocity $v_0$ and the final velocity $v$. If you add them and divide by 2, you get another expression for the average velocity:

$ \bar {v} = \dfrac{v_0 + v}{2} $.

Substituting this into the equation above,

$\boxed{\Delta x=\dfrac{1}{2}(v_0+v)t}$.

This is the second kinematic equation.

Now substitute $v$ from the first kinematic equation into the second kinematic equation:

$\Delta x=\dfrac{1}{2}(v_0+v_0+at)t$.

Simplifying,

$\boxed{\Delta x=v_0t+\dfrac{1}{2}at^2}$.

This is the third kinematic equation.

Now take the first kinematic equation and solve for $t$ to get

$t = \dfrac{v-v_0}{a}$.

Substitute this $t$ into the second kinematic equation to get

$\Delta x=\dfrac{1}{2}(v_0+v)\, \dfrac{v-v_0}{a}$
$\phantom{\Delta x}=\dfrac{1}{2a}(v^2-v_0^2)$.

[Here I have used the identity $(A+B)(A-B)=A^2-B^2$.]

Solving for $v^2$,

$\boxed{v^2 = v_0^2 + 2a \Delta x}$

This is the fourth kinematic equation.