Just like momentum, energy is a conserved quantity. This fact cannot be proven any more than Newton's axioms, but it corresponds to our everyday experience.
Conservation of energy means that the sum of all energy forms occurring in a closed system must be constant and must not change over time.
However, different forms of energy can be converted into one another. This can be clearly illustrated using the example of a falling body or an inclined plane. If the body was initially at a height $h$ above a certain point and is subsequently accelerated by gravity, then upon reaching this point, all of its potential energy has been converted into kinetic energy, i.e., the following applies:
\begin{equation}
\frac{1}{2}mv^2 = mgh
\end{equation}
By rearranging for $v$, we obtain a simple way to determine the final velocity:
\begin{equation}
{v = \sqrt{2gh}}
\end{equation}
This depends solely on the difference in altitude. The relationship shown can also be derived without energy conservation using the corresponding equations from kinematics (see vertical throw). However, the method presented here is more elegant and significantly shorter. A further advantage of using energy conservation is that the exact causes and processes leading to the calculated final velocity do not need to be known, as long as the law of conservation of energy holds.
The law of conservation of energy can be demonstrated using a long string pendulum suspended from the ceiling, to which a heavy ball is attached as a weight. A test subject stands with their back against a wall and holds the ball to the tip of their nose. After releasing it and swinging back, the ball will reach the same maximum position due to conservation of energy. However, taking frictional losses into account, the achievable height is lower.