Potential energy just means how much work that force can perform. You can think of it as "latent work" or "work that has not yet been done".
A ball with mass
m at a height
h above the surface of the Earth (
h much smaller than Earth radius) has the potential energy
mgh. This means that the force of gravity (
mg) can perform work equal to
mgh on that ball.
According to the work-energy theorem (as written above by vanhees1), work performed on an object is equal to the change of the kinetic energy of that object.
Here is a simpler proof of that theorem, considering the special case of a constant force (which is applicable to gravitational force close to the surface of the Earth).
Work:
W = Fh = mgh.
But, because
F = ma we can write the work as
W = mah. Note that because the force
mg is constant, we also have constant acceleration
a = g. So, we can use the basic formulas
v = v0 +
at and
s = (
v + v0)
t/2.
Now, use that
a = g and that the displacement
s is equal to
h.
## W = mah = m \cdot \dfrac{v-v_0}{t} \cdot \dfrac{(v+v_0)t}{2} = \dfrac{mv^2}{2} - \dfrac{mv_0{}^2}{2}## which is the change in the kinetic energy ##E_\text{k,final} - E_\text{k,initial}##.
@vanhees71 you probably missed the "B" tag :)