State and establish the principle of conservation of mechanical energy.

(N/A) The principle of conservation of mechanical energy states that if only conservative forces (like gravitational force) do work on a system,the total mechanical energy (sum of kinetic energy $K$ and potential energy $U$) remains constant.
Proof:
Consider an object of mass $m$ falling freely under gravity from a height $h$ above the ground.
$1$. At height $h$ (point $A$): The velocity $v = 0$. Kinetic energy $K = 0$. Potential energy $U = mgh$. Total energy $E = K + U = mgh$.
$2$. At a distance $x$ below $A$ (point $B$): The object has fallen a distance $x$,so its height is $h - x$. Using $v^2 = u^2 + 2as$,$v^2 = 0 + 2gx = 2gx$. Kinetic energy $K = \frac{1}{2}mv^2 = mgx$. Potential energy $U = mg(h - x)$. Total energy $E = K + U = mgx + mgh - mgx = mgh$.
$3$. At the ground (point $C$): The object has fallen a distance $h$. Using $v^2 = u^2 + 2as$,$v^2 = 0 + 2gh = 2gh$. Kinetic energy $K = \frac{1}{2}mv^2 = mgh$. Potential energy $U = 0$. Total energy $E = K + U = mgh + 0 = mgh$.
Since the total mechanical energy is $mgh$ at all points,the principle is established.