Conservation of Energy

Energy Accounting & Conservation

Conservation laws are physics’ accounting tricks: instead of following every push and pull through time, you compare two snapshots—energy before and energy after. In an isolated system where only conservative forces act, total mechanical energy is conserved: $\Delta E = \Delta KE + \Delta PE = 0$.

Energy conservation equation: $\frac{1}{2}mv_i^2 + PE_i = \frac{1}{2}mv_f^2 + PE_f$.

Problem-solving tip: Energy methods bypass acceleration and time. If a problem asks for speed at a given position, try energy conservation before $F=ma$.

Conservative vs Non-Conservative

A force is conservative if the work it does is path-independent, depending only on start and end points. Equivalently, the work around any closed loop is zero. Gravity and springs are conservative; friction and air resistance are not.

When non-conservative forces are present: $E_f = E_i + W_{\text{nc}}$, where $W_{\text{nc}} < 0$ for friction as energy turns to heat.

Common pitfall: "Energy is conserved" does not mean "mechanical energy is conserved." Friction quietly converts $KE$ into thermal energy. Total energy survives, but mechanical books stop balancing when sliding friction appears.