Physics quiz question: CERN protons

Physics 205B Quiz 7, Fall Semester 2010
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 26.7.

The CERN Large Hadron Collider in Switzerland accelerates protons up to a speed corresponding to a Lorentz factor gamma of 3.000e3, as measured from a stationary reference frame (http://faculty.tamu-commerce.edu/cbertulani/cab/papers/7420.pdf). What is the speed of each proton?
(A) 0.9996667c.
(B) 0.9998333c.
(C) 0.9999998889c.
(D) 0.9999999444c.

Correct answer: (D)

The Lorentz factor gamma, in terms of the speed v is given by:

gamma = sqrt(1/(1 - (v/c)^2));

which can be algebraically manipulated to express v in terms of gamma:

gamma^2 = 1/(1 - (v/c)^2);

1 - (v/c)^2 = 1/(gamma^2);

(v/c)^2 = 1 - 1/(gamma^2);

v = c*sqrt(1 - 1/(gamma^2));

such that with gamma = 3.000e3;

v = c*sqrt(1 - 1/(3.000e3)^2) = c*sqrt(1 - 0.0000001111) = c*sqrt(0.9999998889) = 0.9999999444*c.

Response (A) is 1-(1/gamma); response (B) is sqrt(1-(1/gamma)); response (C) is 1-(1/gamma^2).

Student responses
Section 70856
(A) : 1 student
(B) : 1 student
(C) : 3 students
(D) : 6 students

Success level: 54%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.67