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hw1 - 1-5 This is a case of dilation T = T in this problem...

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1-5 This is a case of dilation. T = " # T in this problem with the proper time " T = T 0 T = 1 " v c # $ % & ( 2 ) * + + , - . . " 1 2 T 0 / v c = 1 " T 0 T # $ % & ( 2 ) * + + , - . . 1 2 ; in this case T = 2 T 0 , v = 1 " L 0 2 L 0 # $ % & ( 2 ) * + , + - . + / + 1 2 = 1 " 1 4 0 1 2 3 4 5 # $ % & ( 1 2 therefore v = 0.866 c . 1-6 This is a case of length contraction. L = " L # in this problem the proper length " L = L 0 , L = 1 " v 2 c 2 # $ % & ( " 1 2 L 0 ) v = c 1 " L L 0 * + , - . / 2 # $ % % & ( ( 1 2 ; in this case L = L 0 2 , v = 1 " L 0 2 L 0 # $ % & ( 2 ) * + , + - . + / + 1 2 = 1 " 1 4 0 1 2 3 4 5 # $ % & ( 1 2 therefore v = 0.866 c . 1-7 The problem is solved by using time dilation. This is also a case of v << c so the binomial expansion is used " t = # " $ t % 1 + v 2 2 c 2 & ( ) * + " $ t , " t # " $ t = v 2 " $ t 2 c 2 ; v = 2 c 2 " t # " $ t ( ) " $ t % & ( ) * 1 2 ; " t = 24 h day ( ) 3 600 s h ( ) = 86 400 s ; " t = " # t $ 1 = 86 399 s ; v = 2 86 400 s " 86 399 s ( ) 86 399 s # $ % % & ( ( 1 2 = 0.004 8 c = 1.44 ) 10 6 m s . 1-8 L = " L # 1 " = L # L = 1 $ v 2 c 2 % & ( ) * 1 2 v = c 1 $ L # L + , - . / 0 2 % & ( ) * * 1 2 = c 1 $ 75 100 + , - . / 0 2 % & ( ) * * 1 2 = 0.661 c 1-10 (a) " = # $ " where " = v c and " # = 1 $ % 2 ( ) $ 1 2 = # & 1 $ v 2 c 2 ( ) * + , $ 1 2 = 2.6 - 10 $ 8 s ( ) 1 $ 0.95 ( ) 2 [ ] $ 1 2 = 8.33 - 10 $ 8 s (b) d = v " = 0.95 ( ) 3 # 10 8 ( ) 8.33 # 10 8 s ( ) = 24 m 1-12 (a) 70 beats/min or " # t = 1 70 min
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