Start with one of our constant acceleration equations, applied in the y-direction:

v_y² = v_oy² + 2a_y Δy

Just for fun, multiply through by m/2:

½m v_y² = ½m v_oy² + ma_y Δy

This gives us two kinetic energy expressions, the final kinetic energy and the initial kinetic energy. These are connected by the third term in the equation.

Consider a free-fall example. A ball is thrown straight up from an initial height of y=0 and reaches a maximum height y=h before falling back down to y=0.

For the going-up part of the trip, the final velocity is zero. We get:

0 = ½m v_oy² - mgh

This gives mgh = ½m v_oy²

For the going-down part of the trip it's the initial velocity that is zero, so:

½m v_y² = mgh

This tells us that the final velocity on the way down equals the initial velocity on the way up, but we knew that.

This analysis tells us something new, though. The kinetic energy decreases on the way up, and increases back to the original amount on the way down. Where does the energy go? What does this "mgh" thing represent. It's certainly some kind of energy - what should we call it?

A good name for it is potential energy, specifically gravitational potential energy. For an object moving under the influence of gravity alone we have this transformation of energy, either from potential to kinetic or vice versa.

Kinetic energy is energy associated with motion.
Potential energy is energy associated with position.