33_InstSolManual_PC - RELATIVITY E XERCISES Section 33.2...

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Unformatted text preview: RELATIVITY E XERCISES Section 33.2 Matter, Motion, and Ether 13. I NTERPRET In this problem we are asked to take wind speed into consideration to calculate the travel time of an airplane. D EVELOP Since the velocities are small compared to c , we can use the non-relativistic Galilean transformation of velocities in Equation 3.7, , u u v = + r r r where u r is the velocity relative to the ground ( S ), u r is that relative to the air ( ), S and v is that of S relative to S (in this case, the wind velocity). We used a notation consistent with that in Equations 33.5a and 33.5b. E VALUATE (a) If v = r (no wind), then u u = r r (ground speed equals air speed), and the round-trip travel time is 2 2(1800 km) 4.5 h 800 km/h a d t u = = = (b) If v r is perpendicular to , u r then 2 2 2 , u u v = + or 2 2 2 2 (800 km/h) (130 km/h) 789 km/h u u v =- =- = and the round-trip travel time is 2 / 4.56 h. b t d u = = (c) If v r is parallel or anti-parallel to u r on alternate legs of the round-trip, then u u v = and the travel time is (see Equation 33.2, but with c replaced by ) u 1800 km 1800 km 4.62 h 800 km/h 130 km/h 800 km/h 130 km/h c d d t u v u v = + = + = +- +- A SSESS We find , a b c t t t < < as mentioned in the paragraph following Equation 33.2. 14. The difference between Equations 33.2 and 33.1 is || 2 2 2 2 2 2 2 2 2 2 2 1 1 1 / 1 / cL L L t t t c c v v c c v v c =- =- =- ---- and 8 2 / 2(11 m)/(3 10 m/s) 73.3 ns L c = = for the given interferometer. If 2 2 / 1, v c << we can expand the denominators (Appendix A) to obtain 2 2 2 2 2 2 (2 / )[1 / (1 /2 )] (2 / )( /2 ). t L c v c v c L c v c +- + = (a) For the Earths orbital speed (30 km/s), 2 2 8 / 10 , v c- = so that 8 16 1 2 73.3 ns( 10 ) 3.67 10 s t-- = = (a fraction of the period of visible light). (b) If 2 2 4 12 / 10 , 3.67 10 s v c t-- = = (a few thousand periods of visible light). (c) At relativistic speeds, we use the exact expression for . t If 2 2 2 2 1 4 / , 1/ 1 / 2/ 3, v c v c = =- = and 4 3 73.3 ns( 2/ 3) 13.1 ns. t =- = (d) If / 0.99, v c = 1/ 1 0.98 7.09, =- = and (2 / ) ( 1) 3.17 s. t L c =- = Section 33.4 Space and Time in Relativity 15. I NTERPRET This problem is about the distance between two stars, as seen by a spaceship moving at relativistic speed. D EVELOP The distance between stars at rest in system S appears Lorentz-contracted in the spaceships system , S according to Equation 33.4: 33.1 33 33.2 Chapter 33 2 2 1 / x x v c = - E VALUATE With 50 ly x = and 0.75 , v c = we get 2 2 2 1 / (50 ly) 1 (0.75) 33.1 ly x x v c = - =- = A SSESS The distance appears to be shortened or contracted as observed by the spaceship. Note that length contraction occurs only along the direction of motion....
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This note was uploaded on 09/12/2009 for the course PH PH101 taught by Professor Prof.kim during the Spring '09 term at Korea Advanced Institute of Science and Technology.

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33_InstSolManual_PC - RELATIVITY E XERCISES Section 33.2...

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