From The Mechanical Universe The Earth in its headlong dash

From The Mechanical Universe: \"The Earth, in its headlong dash around the Sun, is shortened by no more than the length of one blade of grass.\" Calculate the speed of the Earth around the Sun. (You may find Other Useful Data on the inside front cover helpful.) In what frame of reference would an observer see the Earth \"shortened\"? Estimate how much shorter or narrower the Earth will appear to this observer. Stan, standing in the station, sees Mack\'s ship passing by in a time interval of Delta t = 3.57 mu s. If Mack\'s ship has a proper length L_0 = 220 m (i.e., in Mack\'s own frame), Stan sees Mack\'s ship moving at a speed v = L_)/gammaDelta t. Find Mack\'s speed in Stan\'s frame, relative to the speed of light. We are located approximately 23,000 ly from the center of the Milky Way Galaxy. What speed relative to the speed of light,v/c would be required to travel this distance in exactly 30 y in your frame of reference? (Keep extra sig figs!) What value of y would be required? What distance (in ly) would you perceive the Galaxy to pass by you during that trip?

Solution

18.

Sun to earth distance = 1.5 e+11 m

we assume the earth orbit is circular and the period of the orbit is 1Yr

distance traveled in 1 yr = 2*pi*1.5e+11 m

time t = 365*24*3600 s

speed of earth orbiting v = 2*pi*1.5e+11 m/365*24*3600 s

                                         = 2.99e+4 m/s

Earth is orbiting around the sun at a speed comparable to light 3.0e+8 m and we can consider the motin as relativistic.

In the frame at rest relative to earth i.e. Sun\'s frame of reference an observer would see earth is shortened relativistically

Observed length Lo = Lp*sqrt(1-v²/c²) = L_p(1-v²/2c²) = L_p(1-0.5e-8)

earth is shortened by a factor of

Lp-Lo   = Lp*0.5e-8

(Lp-Lo)/Lp   = 0.5e-8 of its proper length

earth proper radius is taken a 6.5 e+6 m

shortend by 2*6.5e+6*0.5e-8 m= 6.5 cm