An unstable particle lasts only a limited proper time. Since it moves no faster

Question

An unstable particle lasts only a limited proper time. Since it moves no faster than the speed of light, you might think that this means that it cannot go an arbitrary distance. This is not correct. (a) If a particle lasts for proper time T and is moving at velocity v, find a formula for how far it goes before it decays. Use v for the velocity, tau for T and gamma for the Lorentz factor y. d= (b) Using the formula you found in part (a), together with the formula for momentum of a particle, find a simple formula for the momentum p in terms of the mass m, the proper time tau and the distance traveled d. P = (c) A typical neutron with mass 1.675 x 10^-27 kg might last for only Tau = 907 s before decaying. If a neutron with this proper lifetime were to make it here from The Large Magellenic Cloud at a distance of 157 kly, what would be the minimum momentum it would have to have? (1 kly = 1000 ly) P = kg*m/s

Explanation / Answer

a) so time in this frame = gamma tau

d = v*t = gamma v tau

b) p = gamma m v

but from a, v = d/( gamma tau)

p = gamma m d/( gamma t) = m d/tau

c)
1 light year = 9.46E15 m
p = 1.675E-27*(157.0E3*9.46E15)/907 = 2.74E-9 kg m/s

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