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

..
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 =

.
y

and the initial velocity v(0)=u. From these equations we can deduce the energy equation.

..
y

= g

where y is measured upwards. Integrating wrt t

.
y

= v = u + gt

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

y = ut + 1
2
gt²

Eliminating
t= v - u
g

we get

y = u(v - u)
g
+ g
2
(v - u)²
g²

which simplifies to

2gy = v² - u².

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 = 1
2
mv² - 1
2
mu²