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UNM Physics 262, Fall 2006
SAMPLE Midterm Exam 2: Relativity
Name and/or CPS number:
Show all your work for full credit. Remember that quantities have units and vectors have components (or magnitude and direction). ASK if anything seems unclear.
CALCULATORS AND CELL PHONES ARE PROHIBITED.
USE OF THESE WILL RESULT IN A ZERO FOR THE EXAM.
Keep any factors of π, e,
2 , etc. in your answers.
You may use a single 8 .5” × 11” paper containing notes you have prepared ahead of time to assist you.
Apportion your time sensibly. Spend about 1012 minutes per problem.
Please put a box around your nal answers.
Problem 1:
Problem 2:
Problem 3:
Problem 4:
- Short answer [25 points] The questions below should be answered with no more than ve lines of text and no calculations. Please be brief and to the point.
[5] a) De ne an inertial reference frame. Give an example of a non-inertial reference frame.
[5] b) What does it mean for two spacetime events to be spacelike separated? Timelike separated? Lightlike separated?
- Spacetime geometry [25 points] By chance, the twins Alice and Bob have enrolled in Physics 262. Inspired by the course, Bob has devised a clever strategy for preserving his youth. (Alice is more sensible.) Bob has attached himself to a spring so that his x-coordinate oscillates according to
x(t) = x 0 sin ωt.
The product x 0 ω has units of velocity, so de ne β 0 ≡ x 0 ω/c. Excited by the prospects of preserving his youth, but wary of testing the limits of his mechanics, Bob sets up the spring so that β 0 1 , and begins to oscillate. Alice watches, rolling her eyes, from her inertial reference frame.
[8] a) Write down an expression relating the dierential passage of time in Alice's IRF (dt) to the dierential passage of Bob's proper time (dτ ). Simplify the relation to the point where β 0 is the only variable in the expression.
[9] b) Integrate this expression to nd how much younger Bob is than Alice after each oscillation, that is, each time Bob returns to x = 0. [Hint: Use β 0 1 and the binomial expansion to make the calculation possible.]
[8] c) For β 0 = 1/ 4 , how long would Bob have to oscillate to be one year younger than Alice?
- Relativistic Dynamics [25 points] A particle is measured in a certain inertial reference frame to have a total energy of 5 GeV and a momentum of 3 GeV/c (i.e., cp, which has the dimensions of energy, is equal to 3 GeV).
[6] a) What is the mass of the particle, in GeV/c^2?
[6] b) What is the velocity of the particle, v/c?
[6] c) What is the energy of the particle in an IRF in which the particle's momentum is 4 GeV/c?
[6] d) What is the kinetic energy of the particle in this new IRF?
[1] e) What is the maximum momentum this particle can have, according to the limits set by special relativity?
