Physics quiz question: water speed flowing through narrowing pipe

Physics 205A Quiz 5, spring semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 9.50

Water at point [1] flows with a speed of 0.75 m/s through a pipe of 0.20 m inner radius. The pipe at point [2] tapers down to an inner radius of 0.050 m. Assume ideal fluid flow. The speed of the water at point [2] is:
(A) 3.0 m/s.
(B) 4.0 m/s.
(C) 12 m/s.
(D) 15 m/s.

Correct answer (highlight to unhide): (C)

From applying the continuity equation:

A₁·v₁ = A₂·v₂,

the water speed at point [2] is greater than at point [1] (v₂ > v₁), because the cross-sectional area at point [2] is smaller than at point [1] (A₂ < A₁), where the cross-sectional areas are given by:

A₁ = π·r₁²;
A₂ = π·r₂².

Then solving for speed of the water at point [2]:

v₂ = A₁·v₁/A₂,

v₂ = π·r₁²·v₁/π·r₂²,

v₂ = (r₁/r₂)²·v₁ = ((0.20 m)/(0.050 m))²·(0.75 m/s) = 12 m/s.

Response (A) is (r₁/r₂)·(0.75 m/s); response (B) is (r₁/r₂); response (D) is (v₁/r₂).

Student responses
Sections 30880, 30881
(A) : 16 students
(B) : 2 students
(C) : 21 students
(D) : 0 students

"Difficulty level": 51%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.80