To determine the lifetime of one particle as observed by the other particle, we need to apply time dilation, which is a consequence of special relativity. Time dilation states that the time experienced by an object moving relative to an observer will appear dilated or stretched compared to the time experienced in its rest frame.
The formula for time dilation is given by:
t' = t / γ
where t' is the observed time, t is the proper time (time experienced in the particle's rest frame), and γ (gamma) is the Lorentz factor, defined as:
γ = 1 / sqrt(1 - (v^2 / c^2))
In this case, both particles, a and b, are moving towards each other with a speed of 0.8c (where c is the speed of light). Let's calculate the Lorentz factor γ first:
γ = 1 / sqrt(1 - (0.8c)^2 / c^2) = 1 / sqrt(1 - 0.64) = 1 / sqrt(0.36) = 1 / 0.6 = 1.67
Now, let's calculate the observed time for particle a as observed by particle b:
t'_a = t_a / γ
where t_a is the proper lifetime of particle a (2 ms):
t'_a = 2 ms / 1.67 ≈ 1.20 ms
Similarly, the observed time for particle b as observed by particle a will also be approximately 1.20 ms.
Therefore, the lifetime of one particle as observed by the other particle is approximately 1.20 ms, not 9.11 ms.