Chapter 39 HW Solutions ----------------------- 39-1: The ratio is just the gamma expansion factor Ratio = Tau_lab/Tau_rest = gamma*Tau_rest/Tau_rest = gamma = = 1/sqrt(1-(v/c)^2) = 1/sqrt(1 -(.9)^2) = 2.29 ---- 39-4: Find the distance to Alpha Centauri- D = (4.3LY)(365days/yr)(24hr/day)(3600s/hr)3x10^8m/s The time wrt earth- T_earth = (D/v) = time for voyage w.r.t. earth By going very fast the clocks on earth will run slower and- T_ship = T_earth/gamma = (D/v)*sqrt(1-(v/c)^2) Solving for beta=v/c c*T_ship = (D/beta)*sqrt(1-beta^2) beta = v/c = sqrt[ D^2/((c*T_ship)^2 + D^2) ]= 0.9999998 --------- 39-8 Initial calculations- L_earth = 2.0m and v/c =.9, v = 2.7 x 10^8 m/s, gamma = 2.294157 Find the travel time in the earth frame- Travel time = T_earth = L_earth/v = .74 x 10^(-8) s In the particle's frame the earth clocks run slower by gamma- T_0 = T_earth/gamma T_0 = .32 10^(-8) s -------------- 39-11 Observers will measure a contracted length by gamma gamma = 1/(1-.8^2) = 1.666667 L_obs = L_ship/gamma = 100m/1.666667 = 60 m ----- 39-12 L/L_0 = 1/2 = 1/gamma = sqrt(1-(v/c)^2) v/c = sqrt(1-(1/2)^2) = 0.8660254 39-17 From earth one side of the box looks shorter by gamma (contracted)- V_earth = L_earth* W* H = (30/gamma)cm * 30cm * 30cm = gamma = 1.666667 V_earth = [30^3/1.666667]cm = 1.6x10^4 cm^3 ------------- 39-19 The distance to Sirius as measured by earthlings is- D = (9LY)(365days/yr)(24hr/day)(3600s/hr)3x10^8m/s Sirians aboard ship see this distance as being contracted- D_ship = D/gamma = D*sqrt(1-(v/c)^2) = v*T_ship T_ship=12 yrs Solving for v/c - v/c = 1/sqrt(1+(c*T_ship/D)^2) = 1/sqrt[1+(1+12/9)^2] = .6 --- Here I used c*T_ship = 12 LY 39-24 In the ground frame the two ray-guns fire simultaneously. The travelers first see hole 2' and then some time later hole 1', a distance L'>L apart. I assume they know special relativity and the rest length L=Lo on the ground. (a) The holes are L apart as measured by the marksmen. (b) Passengers measure the distance D' with a tape measure and then compare it to a measure of the distance between ray-runs on the ground which they see as L'=Lo/gamma. They conclude that the distance D' = gamma * Lo. D' = Lo/sqrt(1-(u/v)**2) ------------------ (c) In the lab assume a flash has occurred to signal to both ray-guns when to fire simultaneously at t1=t2=0. Sensors on the ship record the time of impacts at dt1' and dt2'(not simultaneous!) in ship time. The distance between the ray-guns is measured to be L/(2*gamma) by the travelers. dt1' = L/(2*gamma*c) - (u*dt1')/c (see section 39-8) dt2' = L/(2*gamma*c) + (u*dt2')/c Solving for dt1' and dt2' L/(2*gamma*c) L/(2*gamma*c) dt1' = ----------- dt2' = ----------- (1+beta) (1-beta) -------------------------------\ o o \ -----> u /\ 1' /\ 2' / ---*------------------------*--/ <<*>> |<--------- Lo ---------->| o o /\ 1 /\ 2 The second term (u*dt') is the distance the ship moves forward in time dt'. Solving for dt' = dt1'-dt2' using beta=u/c ------------ dt' = (Lo/2*gamma*c)*{[1/(1+beta)]-[1/(1-beta)]} dt' = (Lo/2*gamma*c)*{2 beta/(1-beta^2)} dt' = (Lo*u/c^2)/sqrt(1-beta^2)} (d) The ship observers measure that the distance between ray-gun is just L'= L0*sqrt(1-beta^2) as in part (b). --------------------- (e) D' = Lo*sqrt(1-beta^2) + u dt' = = Lo*sqrt(1-beta^2) + Lo*beta^2/sqrt(1-beta^2) = Lo/sqrt(1-beta^2) ----------------- 39-29 Lorentz transformation: dx' = gamma*(dx-u*dt) (1) dt' = gamma*(dt-beta*x/c) (2) Since the clock is in the S' system dx'= 0 and dt' = t0(proper time) Solving (1) for dt- dt = (dx'/u)/gamma = (dx'/u)*sqrt(1-beta^2) dt = (200m/.8c)*sqrt(1-(.8)^2) = 5 x10^-7 s ----------- 39-31 Vx = (Vx' + u)/(1+Vx'*u/c^2) (a) Vx = ( .7c + .5c)/(1+.5*.7) = +.89c (b) Vx = (-.7c + .5c)/(1-.5*.7) = -.31c 39-36 Vx = (Vx' + u)/(1+Vx'*u/c^2) Vy = Vy'*sqrt(1-beta^2)/(1+Vx'*u/c^2) Vx = u = .85c since Vx' = 0 ------ Vy = (-.8c)*sqrt(1-(.8)^2)/1 = -.421c ------- 39-37 Vx = (Vx' + u)/(1+Vx'*u/c^2) = .963c ------ Vy = Vy'*sqrt(1-beta^2)/(1+Vx'*u/c^2) = .251c ------ V = sqrt(Vx^2 + Vy^2) = .995c ------ with u = .95c Vx' = .95c cos(60deg) = .475c Vy' = .95c sin(60deg) = .823c 39-38 P = Mo*gamma*beta*c, E = Mo*gamma*c^2, E = Mc^2 = T + Mo*c^2 gamma = 1/sqrt(1-beta^2) = 1/sqrt(1-.8^2) = 1.6667 P = (9.11x10^-11kg)(.80)(3x10^8m/s)*(1.6667) = (3.64x10^-22)kg-m/s ------------------- E = (9.11x10^-11kg)(3x10^8m/s)^2*(1.6667) = (1.37x10^-13)J --------------- Moc^2 = (9.11x10^-11kg)(3x10^8m/s)^2 = (8.20x10^-14)J --------------- T = E-Mo*c^2 = (1.37x10^-13)J - (8.20x10^-14)J = (5.50x10^-14)J -------------- 39-42 Mo = 140 MeV To=1.8x10^-8s = half life Tlab = gamma*To where gamma = E/Mo Tlab = E*To/Mo = (400MeV)(1.8x10^-8s)/140 = 5.14x10^-8s ------------ This is the half-life in the lab system. 39-43 P = Mo*gamma*beta*c, E = Mo*gamma*c^2, E = Mc^2 = T + Mo*c^2 gamma = E/Moc^2 = 1500 MeV/939 MeV = 1.60 = 1/sqrt(1-beta^2) 1-beta^2 = (1/1.6)^2 beta = sqrt(1-(1/1.6)^2) = .78 P = beta*E/c = (.78)(1500MeV/c) = 1170 MeV/c 39-46 M = E/c^2 = (1.5x10^19 J)/(3x10^8m/s) = 167 kg ------ 39-51 dt = d/sqrt(1-beta^2) so 1-beta^2 = (d/dt)^2 and beta = sqrt(1-(d/dt)^2) 39-55 (a) In the rest frame of the meson each gamma ray carries off half the total rest mass energy. Eg1 = Eg2 = (135 MeV)/2 = 67.5 MeV (b) Use the Lorentz transformation for energy and momentum to evaluate in the lab. E' = gamma*(E - beta Px*c) E = gamma*(E' + beta Px'*c) Px' = gamma*(Px - beta E/c) Px = gamma*(Px' + beta E'/c) Py' = Py Py = Py' Pz' = Pz Pz = Pz' beta = .9c gamma = sqrt(1-(.9)^2)=2.294 Eg1 = 2.294*[67.5MeV+.9*(67.5MeV/c)] = 294 MeV + direction Eg2 = 2.294*[67.5MeV+.9*(-67.5MeV/c)] = 15.5 MeV - direction 39-59 o Transverse Doppler Shift- Let To be the period of the wave with respect to the source. We observe the period to be longer by gamma. T = gamma*To The observed frequency is f = 1/T = f0*sqrt(1-beta**2) ----------------------------- o Longitudinal Doppler Shift- Observer travels u*(gamma*To)/c between ticks T = gamma*To + (u*gamma*To)/c approaching T = gamma*To - (u*gamma*To)/c receding T = (1+beta)*gamma*To approaching T = (1-beta)*gamma*To receding f = 1/T = [sqrt(1-beta^2)/(1+beta)]*fo approaching f = 1/T = [sqrt(1-beta^2)/(1-beta)]*fo receding f = fo*sqrt(1-beta)/sqrt(1+beta) approaching ------------------------------------------ f = fo*sqrt(1+beta)/sqrt(1-beta) receding -----------------------------------------