Physics quiz question: cold-cranking battery energy

Physics 205B Quiz 4, spring semester 2012
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 18.11(b), Comprehensive Problem 18.99

An Optima Yellowtop® D51 dual purpose car battery[1] is claimed to be able to provide an equivalent[2] 450 A of current at 7.2 V for 30 seconds. How much electric energy is supplied by the battery during this process?
(A) 0.48 J.
(B) 1.1×10² J.
(C) 1.9×10³ J.
(D) 9.7×10⁴ J.

Sources:
[1] http://www.optimabatteries.com/_media/documents/specs/D51_D51R.pdf.
[2] "Cold cranking amperes," http://en.wikipedia.org/wiki/Automotive_battery#Terms_and_ratings.

Correct answer: (D)

Although there are number of equations that could be used to calculate the correct answer, here a purely unit analysis method is used. Given:

450 A = 450 C/s;
7.2 V = 7.2 J/C;
30 s = 30 s.

Seeking an answer in joules, the trial solution is:

? J = (450 C/s)^a * (7.2 J/C)^b * (30 s)^c.

From inspection, the exponents a, b, and c are all unity, such that the units of C and s cancel out:

? J = (450 C/s)*(7.2 J/C)*(30 s) = 97,200 J = 9.7×10⁴ J.

Response (A) is (7.2 V)*(30 s)/(450 A); response (B) is (450 A)*(7.2 V)/(30 s); and response (C) is (450 A)*(30 s)/(7.2 V).

Section 30882
Exam code: quiz04j0uL
(A) : 3 students
(B) : 6 students
(C) : 6 students
(D) : 11 students

Success level: 42%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.65