Here's how the equations of motion are derived:
For a falling particle the gravitational acceleration is given by

\overset{\cdot\cdot}{y} = g

where the 'double dot' denotes differentiation with respect to time

t

and

y

is measured vertically downwards. By integrating this equation we can derive the equations of motion where the velocity at time t is denoted by

v = \overset{\cdot}{y}

and the initial velocity

v (0) = u

. From these equations we can deduce the energy equation.

$\overset{\cdot\cdot}{y} = g$

where $y$ is measured upwards. Integrating wrt $t$

$\overset{\cdot}{y} = v = u + gt$

Integrating a second time and taking $y = 0$ when $t = 0$

$y = ut + \frac{1}{2} {gt}^{2}$

Eliminating $t = \frac{v - u}{g}$ we get

$y = \frac{u (v - u)}{g} + \frac{g}{2} \frac{(v - u)^{2}}{g^{2}}$

which simplifies to

$2 gy = v^{2} - u^{2} .$

This is equivalent to the energy equation for a mass $m$ where the change in potential energy in falling a distance $y$ is equal to the change in kinetic energy given by the equation:

$mgy = \frac{1}{2} {mv}^{2} - \frac{1}{2} {mu}^{2}$