What Is Energy? — Question 4

Question 4

Two routes can have the same endpoint energy

For this introductory problem use the supplied accounting rule $\Delta E=E_{\mathrm{in}}-E_{\mathrm{out}}$, with all quantities in joules. A device goes from a 40 J state to a 75 J state. Route A supplies 50 J and releases an unknown amount. Route B supplies 90 J and releases an unknown amount. There are no other transfers.

1. Find the released energy along each route.

2. Which quantity is fixed by the endpoint states, and which quantities depend on the route?

3. Can endpoint energy records alone determine the incoming energy? Explain using these routes.

Question 4 - Solution

Two routes can have the same endpoint energy

(a) Use one endpoint difference.

The governing relation is $$\boxed{E_f-E_i=E_{\mathrm{in}}-E_{\mathrm{out}}}$$ The stored-energy increase is $\Delta E=75-40=35$ J. Therefore $$\boxed{E_{{\mathrm{out}},A}=50-35=15\ \mathrm J},\qquad\boxed{E_{{\mathrm{out}},B}=90-35=55\ \mathrm J}.$$

(b) Compare the two ledgers.

The common endpoint change fixes only the net transfer: $$\boxed{E_{\mathrm{in}}-E_{\mathrm{out}}=E_f-E_i=35\ \mathrm J}.$$ The two routes have equal net input but different gross input and output.

(c) Show nonuniqueness explicitly.

No. Both 50 J in with 15 J out and 90 J in with 55 J out fit the same endpoints. An independent input or output measurement is needed.