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NoahsArk
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At the suggestion of some of the members here, I am reading the book Space-Time Physics by Wheeler. The last problem in the first chapter states:
"In a given sample of mesons, half will decay in 18 nanoseconds (18 x 10-9)), measured in a reference frame in which the mesons are at rest. Half of them will decay in the next 18 nanoseconds, and so on.
a) In a particle accelerator mesons are produced when a proton beam strikes an aluminum target inside the accelerator. Mesons leave this target with nearly the speed of light. If there were no time stretching and if no mesons were removed from the resulting beam by collisions, what would be the greatest distance from the target at which half of the mesons would remain undecayed.
b) The mesons of interest in a particular experiment have a speed of .9978 that of light. By what factor is the predicted distance from the target for half-decay increased by the time dilation over the previous prediction- that is, by what factor does the dilation effect allow one to increase the separation between the detecting equipment and target."
For the answer to part a) (which is also given in the book but without explanation), I get 5.4 meters. I did this by calculating 18 nanoseconds by .299972458 meters which is the distance light travels in one nanosecond.
As the answer to part b) (which is not given), I get a factor of 11.74 because the particles can travel 11.74 times farther due to the fact that time in the meson frame, which is moving at .9978, is 11.74 times less than the elapsed time in the lab frame. Am I correct?
I tried to solve this by figuring out what the space time interval was between the creation of the mesons and the moment of their first half life. I took 5.42 - 5.382 (5.4 times .9978)2) = interval 2. interval2 = .22. Interval (or time elapsed in the meson frame) = .46 meters of time which is 11.74 times less than 5.4 meters of time in the lab frame. Is this correct?
I think this also could've been solved by looking up the gamma factor of .9978c. The reason I solved it this way instead is because the book hasn't yet gotten into this yet. They teach how to do it based on the equation for the space time interval. Could these two methods of solving the problem be compared to figuring out the side of a right triangle using the Pythagorean theorem, where we need to know the other two sides' lengths, vs. figuring out the side of a right triangle using trigonometry where we need to only know the length of one other side and the angle (where angle is comparable to speed in a special relativity problem)?
"In a given sample of mesons, half will decay in 18 nanoseconds (18 x 10-9)), measured in a reference frame in which the mesons are at rest. Half of them will decay in the next 18 nanoseconds, and so on.
a) In a particle accelerator mesons are produced when a proton beam strikes an aluminum target inside the accelerator. Mesons leave this target with nearly the speed of light. If there were no time stretching and if no mesons were removed from the resulting beam by collisions, what would be the greatest distance from the target at which half of the mesons would remain undecayed.
b) The mesons of interest in a particular experiment have a speed of .9978 that of light. By what factor is the predicted distance from the target for half-decay increased by the time dilation over the previous prediction- that is, by what factor does the dilation effect allow one to increase the separation between the detecting equipment and target."
For the answer to part a) (which is also given in the book but without explanation), I get 5.4 meters. I did this by calculating 18 nanoseconds by .299972458 meters which is the distance light travels in one nanosecond.
As the answer to part b) (which is not given), I get a factor of 11.74 because the particles can travel 11.74 times farther due to the fact that time in the meson frame, which is moving at .9978, is 11.74 times less than the elapsed time in the lab frame. Am I correct?
I tried to solve this by figuring out what the space time interval was between the creation of the mesons and the moment of their first half life. I took 5.42 - 5.382 (5.4 times .9978)2) = interval 2. interval2 = .22. Interval (or time elapsed in the meson frame) = .46 meters of time which is 11.74 times less than 5.4 meters of time in the lab frame. Is this correct?
I think this also could've been solved by looking up the gamma factor of .9978c. The reason I solved it this way instead is because the book hasn't yet gotten into this yet. They teach how to do it based on the equation for the space time interval. Could these two methods of solving the problem be compared to figuring out the side of a right triangle using the Pythagorean theorem, where we need to know the other two sides' lengths, vs. figuring out the side of a right triangle using trigonometry where we need to only know the length of one other side and the angle (where angle is comparable to speed in a special relativity problem)?