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Unformatted text preview: Chapter 37 p.1 CHAPTER 37 Special Theory of Relativity 1. ( a ) [1 ( v / c ) 2 ] 1/2 = {1 [(20,000 m/s)/(3.00 10 8 m/s)] 2 } 1/2 = 1.00 . ( b ) [1 ( v / c ) 2 ] 1/2 = [1 (0.0100) 2 ] 1/2 = 0.99995 . ( c ) [1 ( v / c ) 2 ] 1/2 = [1 (0.100) 2 ] 1/2 = 0.995 . ( d ) [1 ( v / c ) 2 ] 1/2 = [1 (0.900) 2 ] 1/2 = 0.436 . ( e ) [1 ( v / c ) 2 ] 1/2 = [1 (0.990) 2 ] 1/2 = 0.141 . ( f ) [1 ( v / c ) 2 ] 1/2 = [1 (0.999) 2 ] 1/2 = 0.0447 . 2. You measure the contracted length. We find the rest length from L = L [1 ( v / c ) 2 ] 1/2 ; 28.2 m = L [1 (0.750) 2 ] 1/2 , which gives L = 42.6 m . 3. We find the lifetime at rest from ? t = ? t /[1 ( v 2 / c 2 )] 1/2 ; 4.76 10 6 s = ? t / {1 [(2.70 10 8 m/s)/(3.00 10 8 m/s)] 2 } 1/2 , which gives ? t = 2.07 10 6 s . 4. You measure the contracted length: L = L [1 ( v / c ) 2 ] 1/2 = (100 ly){1 [(2.50 10 8 m/s)/(3.00 10 8 m/s)] 2 } 1/2 = 55.3 ly . 5. We determine the speed from the time dilation: ? t = ? t /[1 ( v 2 / c 2 )] 1/2 ; 4.10 10 8 s = (2.60 10 8 s)/[1 ( v / c ) 2 ] 1/2 , which gives v = 0.773 c . 6. We determine the speed from the length contraction: L = L [1 ( v / c ) 2 ] 1/2 ; 25 ly = (75 ly)[1 ( v / c ) 2 ] 1/2 , which gives v = 0.94 c . 7. For a 1.00 per cent change, the factor in the expressions for time dilation and length contraction must equal 1 0.0100 = 0.9900: [1 ( v / c ) 2 ] 1/2 = 0.9900, which gives v = 0.141 c . 8. In the Earth frame, the clock on the Enterprise will run slower. ( a ) We find the elapsed time on the ship from ? t = ? t /[1 ( v 2 / c 2 )] 1/2 ; 5.0 yr = ? t /[1 (0.84) 2 ] 1/2 , which gives ? t = 2.7 yr . ( b ) We find the elapsed time on the Earth from ? t = ? t /[1 ( v 2 / c 2 )] 1/2 = (5.0 yr)/[1 (0.84) 2 ] 1/2 = 9.2 yr . 9. ( a ) To an observer on Earth, 95.0 ly is the rest length, so the time will be t Earth = L / v = (95.0 ly)/0.960 c = 99.0 yr . ( b ) We find the dilated time on the spacecraft from ? t = ? t /[1 ( v 2 / c 2 )] 1/2 ; 99.0 yr = ? t /[1 (0.960) 2 ] 1/2 , which gives ? t = 27.7 yr ....
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 Spring '08
 ROSS
 Energy, Mass, Theory Of Relativity, Special Relativity, pz, /s

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