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Proper time, denoted as \(\tau\), is a fundamental concept in relativity. It represents the time interval experienced by an object moving along with the particle, meaning it is the time measured in the particle's own rest frame. In our exercise, the proper lifetime of a particle is asked, which is the time that would be measured by an observer moving with the particle, not affected by the effects of relative motion.

To calculate the proper time, we need to use the time dilation formula, which relates the time measured in an observer's frame (lab frame, in this case) with the proper time. This relationship highlights how clocks moving relative to an observer run more slowly, a core phenomenon of special relativity. Understanding this enables us to grasp why proper time is 'shorter' compared to the dilated time measured by a stationary observer.

The Lorentz factor, symbolized by \(\gamma\), is central to calculations in special relativity because it quantifies the amount of time dilation and length contraction. It is calculated using the formula: \(\gamma = \frac{1}{\sqrt{1-v^2/c^2}}\), where \(v\) is the velocity of the object and \(c\) is the speed of light.

In our problem, with the particle traveling at \(0.960c\), the Lorentz factor is computed to be approximately 3.57. This large value indicates significant relativistic effects due to the high velocity, close to the speed of light. As \(v\) approaches \(c\), \(\gamma\) increases dramatically, showcasing how relativistic effects become pronounced at high speeds.

Time dilation is a fascinating consequence of Einstein's theory of special relativity. It implies that time, as measured in a moving frame of reference, will appear to be slower compared to a stationary observer's frame. In simpler terms, if you're moving fast enough, time literally slows down for you relative to someone who is at rest.

In the problem, time dilation tells us that while the lab observer sees the particle as having a longer lifetime, the proper time (lifetime measured in the particle's rest frame) is actually shorter. This is represented mathematically as \(t = \gamma \tau\), linking time \(t\) in the lab's reference frame to the proper time \(\tau\). Here, \(\gamma\) being greater than 1 means \(t\) is expanded compared to \(\tau\).

The rest frame distance refers to the distance as measured by an observer who is at rest relative to the moving object. In the problem, this would be the distance the particle thinks it travels, versus what an external observer might measure.

To determine this distance, denoted \(d_0\), we use its relationship with the proper time: \(d_0 = v \tau\). Here, \(v\) is the velocity, and \(\tau\) is the proper time calculated earlier. By substituting the known values, we can find how far the particle travels in its own frame of reference before decaying. This concept elegantly ties together velocity, time dilation, and the particle's lifetime in understanding motion in relativity.