Solved Problems in Special Relativity Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada Fall 1999; revised 2011 by Malcolm McMillan Given here are solutions to 24 problems in Special Relativity. The solutions were used as a learning-tool for students in the introductory undergraduate course Physics 200 Relativity and Quanta given by Malcolm McMillan at UBC during the 1998 and 1999 Winter Sessions. The solutions were prepared in collaboration with Charles Asman and Adam Monaham who were graduate students in the Department of Physics at that time. The problems are from Chapter 1 Relativity of the course text Modern Physics by Raymond A. Serway, Clement J. Moses and Curt A. Moyer, Saunders College Publishing, 2nd ed., (1997). Standard Inertial Frames We use the standard inertial frames S and S0 which are set up such that the x and x0 axes coincide and the y and y0 axes and z and z0 axes are parallel. Seen from S, S0 moves in the positive x-direction with speed v and, seen from S0, S moves in the negative x0-direction with speed v. Furthermore, it is imagined that in each inertial frame there is an infinite set of recording clocks at rest in the frame and synchronized with each other. Clocks in both frames are set to zero when the origins O and O0 coincide. Time Dilation (\Moving Clocks Run Slow") Problem 1.6, page 45 •At what speed does a clock move if it runs at a rate which is one-half the rate of a clock at rest? Solution We assume that the clock is at rest in S0. As observed by stationary observers in S, the clock moves in the positive x-direction with speed v. Text Eq. (1.30): vx0 t = γ t0 + (1) c2 relates the time t measured in S with the time t0 measured in S0 where 1 γ = (2) p1 − β2 and β = v=c (3) Let ∆t0 be a time interval measured by an observer at rest in S0. (∆t0 is a proper time. ∆t0 is measured when ∆x0 = 0.) 1 Let ∆t be the time interval measured by observers at rest in S. Then ∆t = γ∆t0 (4) and therefore s ∆t0 2 β = 1 − (5) ∆t It follows from Eq. (5) that β = 0:866 when ∆t = 2∆t0. Eq. (4) indicates that the time interval ∆t measured by observers at rest in S is larger than the time interval ∆t0 measured by an observer at rest with respect to the clock. That is, \moving clocks run slow". It is important to note that Eq. (4) relates clock readings on a single clock in S0 with clock readings on two separate clocks in S. "Moving clocks run slow" is illustrated by the LIght Pulse Clock. For this clock, a light pulse is directed along the positive y0 axis and reflected back to its starting point. The traversal time is recorded as ∆t0. In S, the clock moves along the positive x axis with speed v. A stationary observer records the time the light pulse starts and a second stationary observer, farther to the right along the x axis, records the time when the light pulse returns to its starting point. This traversal time is recorded as ∆t. Because of the sideways motion, the light pulse travels farther in S than in S0. Since the speed of light is the same in both frames, it follows that the ∆t is larger than ∆t0. The moving clock runs slow. Length Contraction (\Moving Rods Contract") Problem 1.7, page 45 •At what speed does a meter stick move if its length is observed to shrink to 0.5 m? Solution We assume that the meter stick is at rest in S0. As observed by stationary observers in S, the meter stick moves in the positive x-direction with speed v. Text Eq. (1.25): x0 = γ(x − vt) (6) relates the position x0 measured in S0 with the position x measured in S. Let ∆x0 be the length of the meter stick measured by an observer at rest in S0. (∆x0 is the proper length of the meter stick.) The meter stick is moving with speed v along the x axis in S. To determine its length in S, the positions of the front and back of the meter stick are observed by two stationary observers in S at the same time. The length of the meter stick as measured in S is the distance ∆x between the two stationary observers at ∆t = 0. Then ∆x0 = γ∆x (7) where γ is given by Eq. (2). It follows that s ∆x 2 β = 1 − (8) ∆x0 It follows from Eq. (8) that β = 0:866 when ∆x = ∆x0=2. Eq. (7) indicates that the length ∆x of an object measured by observers at rest in S is smaller than the length ∆x0 measured by an observer at rest with respect to the meter stick. That is, \moving rods contract". It is important to note that Eq. (7) compares an actual length measurement in S0 (a proper length) with a length measurement determined at equal times on two separate clocks in S. 2 Decay of a π Meson: Time Dilation (\Moving Clocks Run Slow") Problem 1.11, page 45 The average lifetime of a π meson in its own frame of reference is 26.0 ns. (This is its proper lifetime.) •If the π meson moves with speed 0:95c with respect to the Earth, what is its lifetime as measured by an observer at rest on Earth? •What is the average distance it travels before decaying as measured by an observer at rest on Earth? Solution We take the S frame to be attached to the Earth and the S0 frame to be the rest frame of the π meson. It follows from Eq. (4) that ∆t = 83:3 ns when ∆t0 = 26:0 ns and v = 0:95c. The average distance travelled before decaying as measured by an observer at rest on Earth is v∆t = 24:0m. Time Dilation for a Slow Moving Object (\Moving Clocks Run Slow") Problem 1.12, page 46 An atomic clock is placed in a jet airplane. The clock measures a time interval of 3600 s when the jet moves with speed 400 m/s. •How much larger a time interval does an identical clock held by an observer at rest on the ground measure? Solution We take the S frame to be attached to the Earth and the S0 frame to be the rest frame of the atomic clock. It follows from Eq. (2) that γ ' 1 + β2=2 (9) and from Eq. (4) that δt = ∆t − ∆t0 ' β2∆t0=2 (10) It follows that δt = 3:2 ns when v = 400 m=s and ∆t0 = 3600 s. Muon Decay: Time Dilation (\Moving Clocks Run Slow") Problem 1.14, page 46 The muon is an unstable particle that spontaneously decays into an electron and two neutrinos. If the number of muons at t = 0 is N0, the number N at time t is −t/τ N = N0e (11) where τ = 2:20µs is the mean lifetime of the muon. Suppose the muons move at speed 0:95c. • What is the observed lifetime of the muons? • How many muons remain after traveling a distance of 3.0 km? Solution 3 We take the S frame to be attached to the Earth and the S0 frame to be the rest frame of the muon. It follows from Eq. (4) that ∆t = 7:046 µs when ∆t0 = 2:2 µs and β = 0:95. A muon at this speed travels 3:0 km in 10:53 µs. After travelling this distance, N muons remain from an initial population of N0 muons where −t/τ −10:53=7:046 N = N0e = N0e = 0:225N0 (12) Length Contraction and Rotation Problem 1.15, page 46 A rod of length L0 moves with speed v along the horizontal direction. The rod makes an angle θ0 with respect to the x0 axis. • Determine the length of the rod as measured by a stationary observer. • Determine the angle θ the rod makes with the x axis. Solution We take the S0 frame to be the rest frame of the rod. 0 0 0 0 A rod of length L0 in S makes an angle θ0 with the x axis. Its projected lengths ∆x and ∆y are 0 ∆x = L0 cos θ0 (13) 0 ∆y = L0 sin θ0 (14) In a frame S in which the rod moves at speed v along the x axis, the projected lengths ∆x and ∆y are given by Eq. (7) and ∆y0 = ∆y (15) which equation follows from text Eq. (1.26): y0 = y (16) The length L of the rod as measured by a stationary observer in S is p 2 2 2 2 1=2 L = (∆x) + (∆y) = L0 1 − β cos θ0 : (17) The rod makes an angle θ with the x axis in S where tan θ = ∆y=∆x = γ tan θ0: (18) The rod in S appears contracted and rotated. See also text Fig. (1.14) on page 19 of the text. Relativistic Doppler Shift Problem 1.18, page 46 • How fast and in what direction must galaxy A be moving if an absorption line found at wavelength 550nm (green) for a stationary galaxy is shifted to 450 nm (blue) (a "blue-shift") for galaxy A? • How fast and in what direction is galaxy B moving if it shows the same line shifted to 700 nm (red) (a "red shift")? Solution 4 Galaxy A is approaching since an absorption line with wavelength 550 nm for a stationary galaxy is shifted to 450 nm.