Physics quiz question: submarine torpedo-eject recoil

Physics 205A Quiz 4, fall semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 7.20

"RC VII class submarine firing torpedoes"
rva1945
youtu.be/1aDDlCHjewc

A submarine of mass 7.69×10⁵ kg[*] and initially at rest ejects a torpedo of mass 1.5×10³ kg[**]. The torpedo has a speed of 10 m/s as it leaves the submarine[***]. Neglect external friction/drag during this brief process. The recoil speed of the submarine is:
(A) 2.0×10^–4 m/s.
(B) 2.0×10^–3 m/s.
(C) 6.7×10^–3 m/s.
(D) 2.0×10^–2 m/s.

[*] wki.pe/German_Type_VII_submarine.
[**] wki.pe/G7a_torpedo.
[***] uboatarchive.net/KTBNotesArmament.htm.

Correct answer (highlight to unhide): (D)

Since there are no significant external impulses on the submarine-torpedo system during this brief process, total momentum is conserved, so:

0 = m₁·(v_f1 – v₀₁) + m₂·(v_f2 – v₀₂),

where for the submarine, m₁ = 7.69×10⁵ kg, v₀₁ = 0, v_f1 = ?; and for the torpedo, m₂ = 1.5×10³ kg, v₀₂ = 0, v_f2 = +10 m/s (in the forward direction). So solving for the final velocity v_f1 of the submarine:

0 = m₁·v_f1 + m₂·v_f2,

–m₁·v_f1 = m₂·v_f2,

v_f1 = –m₂·v_f2/m₁ = –(1.5×10³ kg)·(+10 m/s)/(7.69×10⁵ kg) = –1.950585176×10^–2 m/s,

or to two significant figures, the recoil speed of the submarine is the magnitude of its final velocity: 2.0×10^–2 m/s. (Note even with neglecting drag/friction forces, the recoil speed of the submarine is miniscule compared to the torpedo's launch speed, due to the relatively larger mass of the submarine.)

(Response (A) is m₂/(m₁·v_f2); response (B) is m₂/m₁; response (C) is v_f2/m₂.)

Student responses
Sections 70854, 70855
(A) : 3 students
(B) : 9 students
(C) : 2 students
(D) : 34 students

Success level: 70%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.73

Student responses
Section 72177
(A) : 0 students
(B) : 1 student
(C) : 0 students
(D) : 12 students

Success level: 92%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.20