Conservation of Mechanical Energy

The total mechanical energy of a body is defined as the sum of its kinetic and potential energy at any instant, $E = K + U$. The law of conservation of mechanical energy states that, whenever only conservative forces do work on a body (no friction, air resistance, or other non-conservative force is present, or their effect is deliberately neglected), this total, $E$, remains exactly constant throughout the body's motion, even though $K$ and $U$ individually change continuously, converting back and forth into one another.

Deriving the law. Starting from the work-energy theorem (§5.4), the net work done on a body equals its change in kinetic energy, $W_{\text{net}} = \Delta K$. If the only forces doing work are conservative, then by the definition of potential energy (§5.7), $W_{\text{net}} = -\Delta U$. Combining these two expressions for $W_{\text{net}}$:

Since the change in $K + U$ is exactly zero between any two instants, $K + U$ itself must be constant throughout the motion:

Worked illustration -- free fall. A body of mass $m$ dropped from rest at height $h$ has, at the moment of release, $K_i = 0$ and $U_i = mgh$ (taking the ground as the reference level), so $E_i = mgh$. Just before it strikes the ground, $U_f = 0$ and, since $E$ is conserved, $K_f = mgh$, giving $\tfrac12 mv_f^2 = mgh$ and so $v_f = \sqrt{2gh}$ -- the same familiar result obtained, more laboriously, from the equations of uniformly accelerated motion, but reached here purely from energy bookkeeping, without ever needing to find the body's acceleration or time of fall explicitly.

When non-conservative forces also act, the law must be adjusted: the work-energy theorem still holds exactly ($W_{\text{net}} = \Delta K$), but $W_{\text{net}}$ now includes a non-conservative contribution $W_{nc}$ in addition to the conservative one, so that

$\Delta K + \Delta U = W_{nc}$ …