Q236 · Practice questionComplex familiar5 marks
QUESTION 236 (5 marks)
Two detectors are apart in a laboratory. Unstable particles travel from the first towards the second at and have a mean proper lifetime of . Use .
a)[2 marks]
Determine the travel time between detectors measured by a clock moving with a particle that reaches the second detector.
b)[2 marks]
Determine the mean decay distance in the laboratory.
c)[1 mark]
Does a mean decay distance greater than the detector separation prove that every particle reaches the second detector? Explain.
WORKED SOLUTION
5 marksPractice marking scheme
ANSWER
(a) . (b) . (c) No. A mean lifetime is an average; some particles decay earlier.
Worked solution
(a) The laboratory travel time is . Since , the particle-clock proper time is . Equivalently, the detector separation in the particle frame is , giving .
(b) The laboratory mean lifetime is . Thus the mean decay distance is .
(c) The mean does not assign the same lifetime to every particle. Individual decay times vary, so some particles decay before travelling even though the mean distance is larger.
Equivalent physically justified methods and consistent equivalent units accepted.
Displayed decimals are model answers; accept appropriate significant figures and consistent rounding from stated constants.
Part a: Determine the laboratory travel interval or contracted detector separation.
Part a: Calculate the particle-clock proper travel interval.
Part b: Dilate the proper mean lifetime.
Part b: Multiply by laboratory speed to calculate the mean distance.
Part c: Distinguish a mean lifetime from a guaranteed individual lifetime.
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