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2 2 0 1 \u394 \u394 c v tt \u2212 = \u21d2 2 01 \u239f\u23a0 \u239e\u239c\u239d \u239b \u394 \u394\u2212= t tcv Substitute numerical values and evaluate v: ( ) m/s108.2 s105.7 s106.21m/s10998.2 8 2 8 8 8 ×=\u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b × ×\u2212×= \u2212 \u2212 v 23 \u2022 A meterstick moves with speed 0.80c relative to you in the direction parallel to the stick. (a) Find the length of the stick as measured by you. (b) How long does it take for the stick to pass you? Picture the Problem We can find the measured length L of the meterstick using ( )220 1 cvLL \u2212= and the time it takes to pass you using L = v\u394t. (a) Express the length L of the meterstick in terms of its proper length L0: 2 2 0 1 c vLL \u2212= Substitute numerical values and evaluate L: ( ) ( ) cm608.01m0.1 2 2 =\u2212= c cL Chapter R 1054 (b) Express the time it takes for the meterstick to pass you in terms of its apparent length and speed: v Lt =\u394 Substitute numerical values and evaluate \u394t: ns5.28.0 m60.0 \u394 == c t 24 \u2022 Recall that the half-life is the time it takes for any given amount of unstable particles to decay to half that amount of particles. The proper half-life of a species of charged subatomic particles called pions is 1.80 × 10\u20138 s (See Problem 18 for details on pions.) . Suppose a group of these pions are produced in an accelerator and emerge with a speed of 0.998c. How far do these particles travel in the accelerator\u2019s laboratory before half of them have decayed? Picture the Problem We can express the distance the pions will travel in the laboratory using tvx \u394\u394 = and find their half-life in the accelerator laboratory using ( )20 1\u394\u394 cvtt \u2212= . Express the average distance the pions will travel before decaying in terms of their speed proper mean lifetime: tvx \u394\u394 = Use the time-dilation equation to relate the mean lifetimes of the pions in the accelerator laboratory \u394t to their proper mean lifetime \u394t0: 2 0 1 \u394 \u394 \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212 = c v tt Substitute for t\u394 to obtain: 2 0 1 \u394 \u394 \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212 = c v tvx Substitute numerical values and evaluate \u394x: ( )( ) m2.85 998.01 s1080.1998.0 \u394 2 8 = \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212 ×= \u2212 c c cx 25 \u2022\u2022 [SSM] Your friend, who is the same age as you, travels to the star Alpha Centauri, which is 4.0 light-years away, and returns immediately. He claims that the entire trip took just 6.0 y. What was his speed? Ignore any accelerations of your friend\u2019s spaceship and assume the spaceship traveled at the same speed during the entire trip. Special Relativity 1055 Picture the Problem To calculate the speed in the reference frame of the friend, who is named Ed, we consider each leg of the trip separately. Consider an imaginary stick extending from Earth to Alpha Centauri that is at rest relative to Earth. In Ed\u2019s frame the length of the stick, and thus the distance between Earth and Alpha Centauri, is shortened in accord with the length contraction formula. As three years pass on Ed\u2019s watch Alpha Centauri travels at speed v from its initial location to him. Sketch the situation as it is in your reference frame. The distance between Earth and Alpha C. is the rest length L0 of the stick discussed in Picture the Problem: Alpha C. Your reference frame v Earth Ed moving 0L Sketch the situation as it is in Ed\u2019s reference frame. The distance between Earth and Alpha C. is the length L of the moving stick discussed in Picture the Problem: Alpha C. Your friend Ed's reference frame v Earth Ed at rest L v The two events are Ed leaves Earth and Ed arrive at Alpha Centauri. In Ed\u2019s frame these two events occur at the same place (next to Ed). Thus, the time between those two events \u394t0 is the proper time between the two events. \u394t0 = 3 y The distance L traveled by Alpha Centauri in Ed\u2019s frame during the first three years equals the speed multiplied by the time in Ed\u2019s frame: 0L v t= \u394 The distance between Earth and Alpha Centauri in Ed\u2019s frame is the contracted length of the imaginary stick: 2 0 21 vL L c = \u2212 Equate these expressions for L to obtain: 2 0 021 vL v t c \u2212 = \u394 Chapter R 1056 Substituting numerical values yields: ( ) ( )y 0.31y 0.4 2 2 v c vc =\u2212\u22c5 or c v c v 75.01 2 2 =\u2212 Solve the quadratic equation in v/c ot obtain: 80.0=c v \u21d2 cv 80.0= 26 \u2022\u2022 Two spaceships pass each other traveling in opposite directions. A passenger in ship A knows that her ship is 100 m long. She notes that ship B is moving with a speed of 0.92c relative to A and that the length of B is 36 m. What are the lengths of the two spaceships as measured by a passenger in ship B? Picture the Problem We can use the relationship between the measured length L of the spaceships and their proper lengths L0 to find the lengths of the two spaceships as measured by a passenger in ship B. Relate the measured length LA of ship A to its proper length: 2 A,0A 1 \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212= c vLL Substitute numerical values and evaluate LA: ( ) m3992.01m100 2A =\u239f\u23a0 \u239e\u239c\u239d \u239b\u2212= c cL Relate the proper length L0,B of ship B to its measured length LB: 2 B B,0 1 \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212 = c v LL Substitute numerical values and evaluate B,0L : m92 92.01 m36 2B,0 = \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212 = c c L 27 \u2022\u2022 Supersonic jets achieve maximum speeds of about 3.00 × 10\u20136c. (a) By what percentage would a jet traveling at this speed contract in length? (b) During a time of exactly one year or 3.15 × 107 s on your clock, how much time would elapse on the pilot\u2019s clock? How much time is lost by the pilot\u2019s clock in one year of your time? Assume you are on the ground and the pilot is flying at the specified speed for the entire year. Picture the Problem We can use the relationship between the measured length L of the jet and its proper length to express the fractional change in the length of the Special Relativity 1057 jet traveling at its maximum speed. In Part (b) we can express the elapsed time on the pilot\u2019s clock \u394t0 in terms of the elapsed time \u394t on your clock. (a) Express the fractional change in length of the jet: 00 0 1 L L L LL \u2212=\u2212 Relate L to L0: 2 0 1 \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212= c vLL \u21d2 2 0 1 \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212= c v L L Substitute for 0LL to obtain: 2 2 0 0 11 c v L LL \u2212\u2212=\u2212 (1) Expand 2 2 1 c v\u2212 binomially to obtain: 2 2 4 4 2 2 2 2 2 11 ... 8 3 2 111 2 1 c v c v c v c v \u2212\u2248 ++\u2212=\u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b \u2212 Substituting in equation (1) yields: 2 2 2 2 0 0 2 1 2 111 c v c v L LL =\u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b \u2212\u2212=\u2212 Substitute numerical values and evaluate the jet\u2019s fractional change in length: ( ) %1050.4 1000.3 2 1 10 2 26 0 0 \u2212 \u2212 ×= ×\u2248\u2212 c c L LL (b) Express the elapsed time on the pilot\u2019s clock \u394t0 in terms of the elapsed time \u394t on your clock: t c vt c vtt c vt \u394 2 1 \u394 2 11\u394\u3941\u394 2 2 2 2 2 2 0 2 1 \u2212= \u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b \u2212\u2248\u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b \u2212= where the second term represents the time lost on the pilot\u2019s clock. Substitute numerical values and evaluate the elapsed time on the pilot\u2019s clock in 1 y = 3.15 × 107 s: ( ) ( ) y 00.1s1015.31000.3 2 11\u394 2 11\u394 72 26 2 2 0 =×\u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b ×\u2212=\u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b \u2212= \u2212 c ct c vt Chapter R 1058 Substitute numerical values and evaluate the time lost on the pilot\u2019s clock in 1 y = 3.15 × 107 s: ( ) ( ) s 421 s1015.31000.3 2 1 \u394 2 1 7 2 26 2 2 \u3bc= ××= \u2212 c ct c v 28 \u2022\u2022 The proper mean lifetime of a muon (see Problems 20 and 21 for details) is 2.20 \u3bcs. Consider a muon, created in the Earth\u2019s atmosphere, speeding toward the surface 8.00 km below, at a speed of 0.980c. (a) What is the likelihood that it will survive its trip to ground before decaying? The probability of a muon decaying is given by P = 1\u2212 e\u2212\u394t /\u3c4 , where \u394t is the time interval as measured in the reference frame in question. (b) Calculate this from the point of view of an observer moving with the muon. Show that, from the point of view of