For a car that coasts up a hill and comes to rest (gas runs out, no friction), how do you simplify the work-energy equation $KE_i + PE_i + W_{nc} = KE_f + PE_f$ and what is the sign of $W_{nc}$?

What the situation tells you about energy transfers

The car starts at the bottom moving fast, then coasts uphill and comes to a stop. With no friction and no engine power (it ran out of gas), there is no external energy added or removed; energy just changes form between kinetic and gravitational potential.

Decide which terms are zero

• Nonconservative work: there is no friction and the engine is not providing a driving force, so $$W_{nc}=0.$$
• Final kinetic energy: the car comes to rest, so $$KE_f=0.$$

Write the simplified work-energy equation

Start with $$KE_i + PE_i + W_{nc} = KE_f + PE_f.$$ Cancel the zero terms ($W_{nc}=0$): $$KE_i + PE_i = KE_f + PE_f.$$ Now use $KE_f=0$: $$KE_i + PE_i = PE_f.$$

Interpreting the sign of $W_{nc}$

Because we are ignoring friction and the engine is not doing work on the car, there is no nonconservative work done on the system:

• $W_{nc}$ is neither positive nor negative.
• The correct choice is $W_{nc}=0$.

Quick check (does it make sense?)

Going uphill increases gravitational potential energy, so the car must lose kinetic energy. Ending at rest means all the initial kinetic energy (plus any initial potential, if you chose a different zero level) shows up as increased $PE$.