Chapter 28 ---------- 1). Proper time is measured in the car rest frame. The time measured by an observer on earth is t_earth = t_car / sqrt(1-(v/c)^2) v = c*sqrt(1-(t_car/t_earth)^2) = 0.8*c i.e., the speed of the police car is 0.8 of the speed of light or 0.8*300000000 m/s = 2.4*10^8 m/s 3). In itws own reference frame (where pion is at rest) the pion life time is 2.6*10^(-8) seconds. However in the laboratory frame it appears to be longer due to time dilatation. t = t0 / sqrt(1 - (v/c)^2) t = 2.6*10^(-8) s / sqrt(1 - 0.67^2) = 3.5*10^(-8) s 11). In the rest frame of the UFO, the Earth is moving fast at the speed 0.7c. Thus, the UFO voyagers see the distance between New York and San Francisco much shorter due to the Lorentz contraction: L = Lo * sqrt(1-(v/c)^2) = 4.2*10^6 m * sqrt(1-0.70^2) = 3*10^6 m 12). In the rest frame of the astronaut, the Milky Way is moving at the speed 0.99999c, and he see it much shorter: L = Lo * sqrt(1-(0.99999)^2) = = 1.2*10^5 * 0.00447 = 536 light years. So it will take ~536 years for the astronaut to cross the galaxy. 26). The energy of a moving proton is E = m*c^2 / sqrt(1-(v/c)^2) The proton energy at rest is E_o = m*c^2 From the relation E = 4*E_o we can extract velocity v 4*m*c^2 = m*c^2 /sqrt(1-(v/c)^2) sqrt(1-(v/c)^2) = 0.25 v = 0.94c = 2.8 * 10^8 m/sec