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Homework Statement
A beam on Pions, Kaons and Protons, all with momentum ##\mathrm{P} = 10 \mathrm{GeV}## and negligible angular divergence travels ##100 \mathrm{m}## before hitting a target. What is the required timing resolution of the detector so that pions and kaons can be distinguished with a 10% precision.
Homework Equations
The Attempt at a Solution
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I have calculated the time difference between the pions and kaons as follows:
A particle with mass ##m## and momentum ##p## has velocity ##\beta = \frac{p}{\sqrt{p^2 + m^2}}##. For a path length ##L##, the time of flight ##T = \frac{L}{\beta c}##.
Therefore two particles with different masses, but the same momentum arrive with a time difference of ##T_1 - T_2 = \frac{L}{c}(\sqrt{1 + \frac{m^2_1}{P^2}} - \sqrt{1 + \frac{m^2_2}{p^2}})##
I am unsure of how to take this further because I am confused by what it means to distinguish the two particles with a given precision.
What I am picturing is something like this;
the timing resolution of the detector is basically what determines the width of a 'bin' when counting pions and kaons. I need to make my bin widths small enough so that 10% of a measured arrival time is smaller than the time difference between the two particle's arrivals. Does that make sense?
Thanks for any help you can give!
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