Physics clicker question: instantaneous speed vs. magnitude of instantaneous velocity

Physics 205A, Fall Semester 2008
Cuesta College, San Luis Obispo, CA

Students were asked the following clicker question (Classroom Performance System, einstruction.com) at the middle of their learning cycle, after discussing the distinction between average and instantaneous quantities.

In general, which statement(s) is/are possible?
(A) (Instantaneous) speed > magnitude of (instantaneous) velocity.
(B) (Instantaneous) speed < magnitude of (instantaneous) velocity.
(C) (Instantaneous) speed = magnitude of (instantaneous) velocity.
(D) (More than one of the above choices.)
(E) (I'm lost, and don't know how to answer this.)

Sections 70854, 70855
(A) : 1 student
(B) : 1 student
(C) : 16 students
(D) : 27 students
(E) : 0 students

This question was asked again after displaying the tallied results with the lack of consensus, with the following results. No comments were made by the instructor, in order to see if students were going to be able to discuss and determine the correct answer among themselves.

Sections 70854, 70855
(A) : 0 students
(B) : 0 student
(C) : 25 students
(D) : 20 students
(E) : 0 students

Correct answer: (C)

While not an overwhelming number of correct responses, at least an appreciable shift away from (D), which is an incorrect answer, a point brought up in the whole-class discussion after the second pass.

Instantaneous quantities are evaluated in the limit that the elapsed time interval approaches zero. From a previous discussion in class, average speeds can be greater than the magnitude of average velocities when the trajectory between the initial and final positions doubles back and forth on itself, but instantaneous speeds must be the same as the magnitudes of instantaneous velocities, because in the limit that the elapsed time interval approaches zero, there can be only straight-line travel with no "back-and-forth" between initial and final positions.

In the follow-up discussion, a student asked whether instantaneous speeds can still be greater than magnitudes of instantaneous velocities for something very small, say, like molecules. But even though the brownian motion of molecules would jostle it back and forth over small distances, the nature of the time interval for instantaneous quantities means that it should be made sufficiently small enough that there is no "back-and-forth" between initial and final positions.