When it is stationary, the half-life of a certain subatomic particle is T 0 . Th

Question

When it is stationary, the half-life of a certain subatomic particle is T0. That is, if N of these particles are present at a certain time, then a time T0 later only N/2 particles will be present, assuming the particles are at rest. A beam carrying N such particles per second is created at position x=0 in a high-energy physics laboratory. This beam travels along the x axis at speed v in the laboratory reference frame and it is found that only N/2 particles per second travel in the beam at x=2cT0, where c is the speed of light.

Explanation / Answer

T = gamma T0

v = x/T = (2cT0)/(gamma T0)

==> v = 2c/gamma

gamma = 1/sqrt(1 - (v/c)^2)

==> v = 2c/(1/sqrt(1 - (v/c)^2))

==> v = 2 c sqrt(1 - (v/c)^2))

==> v^2 = 4 c^2 (1 - (v/c)^2)

==> v^2 = 4 c^2 - 4 v^2

==> 5 v^2 = 4 c^2

==> v = 2/(sqrt(5)) c

==> v = 2/2.23607 c

==> v = 0.894 c = 2.68 x 10^8 m/s

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