Conservation of Mechanical Energy is easiest to use as bookkeeping. An energy bar chart shows the energy at the start, the energy that crosses the boundary of your system, and the energy at the end, as bars you can count.
Key idea
Choose the system. Then: energy at the start $+$ work done by outside forces $=$ energy at the end, where the energies are $K$, $U_g$, $U_s$ and thermal energy $E_{th}$. The bars on the left plus the work must equal the bars on the right.
Why it works
Energy can change form inside the system and can cross its boundary as work, but it is never created or destroyed. So the total before, plus whatever enters as work (minus whatever leaves), must equal the total after. Choosing the system decides which forces count as "outside": with Earth inside the system, gravity shows up as $U_g$; with Earth outside, it shows up as work instead. Either way works, as long as you don't count it twice.
Worked examples
Worked example: Pushed cart
A cart with $20\ \text{J}$ of kinetic energy is pushed along a level, frictionless track; the push does $30\ \text{J}$ of work. Find its final $K$.
- System: the cart. Start: $K = 20\ \text{J}$. Work in: $30\ \text{J}$.
- $20 + 30 = K_f$, so $K_f = 50\ \text{J}$.
Answer: $50\ \text{J}$
Worked example: Spring up a ramp
A spring storing $80\ \text{J}$ launches a cart up a frictionless ramp. At one point the cart has $30\ \text{J}$ of kinetic energy. How much $U_g$ does the system have there (zero at the start)?
- System: cart, spring and Earth. No outside work.
- $80 = 30 + U_g$, so $U_g = 50\ \text{J}$.
Answer: $50\ \text{J}$
Worked example: Energy leaving
A skydiver falls at a steady speed and loses $2{,}000\ \text{J}$ of $U_g$. With air drag outside the system (skydiver and Earth), how much work does drag do?
- $K$ stays the same. Start: $U_g$; end: $U_g - 2000\ \text{J}$.
- So the work done by drag is $-2{,}000\ \text{J}$: energy leaves the system as work and ends up warming the air.
Answer: $-2{,}000\ \text{J}$
Common mistake
Leaving out energy that enters or leaves the system as work, such as a push, a cable's pull or air drag. Then the bars won't balance. Check: draw the boundary of your system; every force from outside that moves along a distance must appear as a $W$ bar, positive in or negative out.