What Is Energy? — Question 9

Question 9

A stored energy and its delivery rate answer different questions

Store A can deliver 600 J at a constant rate of 20 J/s. Store B can deliver 450 J at 50 J/s. Ignore all losses. Both start delivering at time zero and stop when empty.

1. Find each delivery duration.

2. How much has each delivered by 5 s, and which lasts longer?

3. When do their cumulative delivered energies first become equal again after time zero?

Question 9 - Solution

A stored energy and its delivery rate answer different questions

(a) Divide capacity by rate.

The governing relation is $$\boxed{E_{\mathrm{delivered}}(t)=\min(Pt,E_0),\qquad t_{\mathrm{empty}}=E_0/P}$$ Capacity divided by constant delivery rate gives the emptying time: $$\boxed{t_A=\frac{600}{20}=30\ \mathrm s},\qquad\boxed{t_B=\frac{450}{50}=9\ \mathrm s}.$$

(b) Use the rate only while energy remains.

At 5 s neither store is empty: $$E_A(5)=20(5)=100\ \mathrm J,\qquad E_B(5)=50(5)=250\ \mathrm J.$$ B has delivered more energy at that instant, while A has the longer delivery duration.

(c) Include the empty-store boundary.

Use the saturation limit of each store: $$E_A(t)=\min(20t,600),\qquad E_B(t)=\min(50t,450)\quad\text{(J; $t$ in s)}.$$ Before 9 s, equality requires $20t=50t$, hence only $t=0$. After B empties, $$20t=450\Longrightarrow\boxed{t=22.5\ \mathrm s}<t_A.$$ This is the only subsequent crossing; A later reaches 600 J.