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Exercise:
The average lifetime of a pion short-lived particle in its rest frame is tO. Calculate the speed of a pion that travels dO during the average lifetime.

Solution:
For an observer in the Earth's rest frame the distance travelled by the pion is given by d v t v gamma tau where tau is the half-life measured in the pion's rest frame. Solving this for v yields: v fracdgamma tau fracdtau fracgamma fracdtau sqrt-beta^ It follows for the dimensionless velocity beta fracvc fracdc tau sqrt-beta^ We solve this for beta^: beta^ leftfracdc tauright^ left-beta^right beta^ left+ leftfracdc tauright^right leftfracdc tauright^ beta^ leftfracc tau^ + d^c tau^right leftfracdc tauright^ beta^ fracd^c tau^ + d^ We finally find for beta beta beF fracdsqrtncc times t^ + d^ resultbeP The pion travels at bepercP of the speed of light.
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Exercise:
The average lifetime of a pion short-lived particle in its rest frame is tO. Calculate the speed of a pion that travels dO during the average lifetime.

Solution:
For an observer in the Earth's rest frame the distance travelled by the pion is given by d v t v gamma tau where tau is the half-life measured in the pion's rest frame. Solving this for v yields: v fracdgamma tau fracdtau fracgamma fracdtau sqrt-beta^ It follows for the dimensionless velocity beta fracvc fracdc tau sqrt-beta^ We solve this for beta^: beta^ leftfracdc tauright^ left-beta^right beta^ left+ leftfracdc tauright^right leftfracdc tauright^ beta^ leftfracc tau^ + d^c tau^right leftfracdc tauright^ beta^ fracd^c tau^ + d^ We finally find for beta beta beF fracdsqrtncc times t^ + d^ resultbeP The pion travels at bepercP of the speed of light.
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Attributes & Decorations
Branches
special relativity
Tags
dilation, pion, relativity, time
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Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
Decoration