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# s101 - Solutions to Homework#10 Chapter 29 12 Picture the...

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Chapter 29 12. Picture the Problem : A pion decays after a proper lifetime of 26 ns. An observer in the laboratory will measure the dilated time when the pion is traveling at 0.99 c . Strategy: The two events to be considered in this case are the creation and the decay of the pion. From an Earth observer’s point of view, these events occur at different places, so she observes the dilated time interval t and an observer in the pion’s frame of reference measures the proper time interval t 0 . Use equation 29-2 to calculate the dilated time in the Earth frame from the speed and proper lifetime of the pion. Calculate the distance as the product of velocity and time. To calculate the distance without relativistic effects, use the rest frame lifetime of the pion. Solution: 1. (a) Calculate the lifetime in the Earth frame: ( 29 8 7 0 2 2 2 2.6 10 s 10 s 1 1 0.99 t t v c - - ∆ = = = 1.8 - - 2. (b) Calculate the distance traveled in the Earth frame: ( 29 ( 29 ( 29 8 8 0 av 2 2 2 0.99 3.00 10 m/s 2.6 10 s 55 m 1 1 0.99 t d v t v v c - = ∆ = = = - - 3. (c) Calculate the distance traveled without relativistic effects: ( 29 ( 29 8 8 av 0 0.99 3.00 10 m/s 2.6 10 s 7.7 m d v t - = ∆ = = Insight: Laboratory experiments with pions confirm that relativistic time dilation does in fact exist. 34. Picture the Problem : The contracted length of a rocket is 9.00 m when it is moving at speed v relative to an observer, and it is 5.00 m when it is traveling at speed 2 v . Strategy: We want to calculate the speed v and the proper length of the rocket. First solve equation 29-3 for the proper length, and then use the proper length to form a ratio from which you can find the speed v . Finally, insert v into equation 29-3 to solve for the proper length L 0 . Solution: 1. Solve equation 29-3 for the proper length of the rocket: 2 2 0 0 2 2 1 1 L L L v c L v c = - = - 2. Use the proper length to create a ratio: ( 29 1 2 0 2 2 2 2 2 2 1 2 2 2 1 1 2 1 1 4 L L L v c v c v c L L v c = = - - - = - 3. Solve for the speed v : ( 29 ( 29 ( 29 ( 29 2 2 1 2 2 2 1 2 1 1 9.00 m 5.00 m 0.433 1 4 1 4 9.00 m 5.00 m L L v c c c L L - - = = = - - 4. Use the equation from step 1 to find L 0 : ( 29 0 2 9.00 m 9.98 m 1

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s101 - Solutions to Homework#10 Chapter 29 12 Picture the...

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