Conservation of Mechanical Energy

The law

When only conservative forces (gravity, spring force) do work on a body, the total mechanical energy — kinetic plus potential — remains constant:

$KE + PE = E_{total} = \text{constant}$

$\frac{1}{2}mv^2 + U = \text{constant}$

As the body moves, KE and PE continuously exchange, but their sum never changes.

Derivation

From the work-energy theorem:

$W_{net} = \Delta KE$

If only conservative forces act, work done by conservative forces = decrease in PE:

$W_{conservative} = -\Delta PE = -(PE_f - PE_i)$

Therefore:

$\Delta KE = -\Delta PE$

$KE_f - KE_i = -(PE_f - PE_i)$

$KE_f + PE_f = KE_i + PE_i$

$\boxed{KE + PE = \text{constant}}$

Applications

Free fall from height $h$:

$\frac{1}{2}mv^2 + mgh = \text{constant}$

At top (rest): $0 + mgh = mgh$

At bottom: $\frac{1}{2}mv^2 + 0 = mgh \implies v = \sqrt{2gh}$

Spring-mass system:

$\frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \text{constant} = \frac{1}{2}kA^2$

where $A$ is the amplitude (maximum displacement). At $x = 0$: all KE. At $x = A$: all PE.

Pendulum:

$\frac{1}{2}mv^2 + mgh = \text{constant}$

At the bottom: maximum KE, minimum PE. At the top of swing: zero KE, maximum PE.

When conservation does NOT hold

If non-conservative forces (friction, air resistance, applied external force) do work, mechanical energy is not conserved:

$\Delta E_{mech} = W_{non-conservative}$

Friction always does negative work — it removes mechanical energy and converts it to heat.

Energy conservation is never violated

Even when mechanical energy is not conserved, total energy (including heat, sound, internal energy) is always conserved. Energy is never created or destroyed — it only changes form.

The conservation of mechanical energy is a restricted version of this universal law, valid only when no energy leaves the mechanical system.