04.1 State what is meant by an inertial frame of reference - AQA - A-Level Physics - Question 4 - 2021 - Paper 7

1. Distance between detectors: The detectors are separated by a distance of 45 m.

   2. Time taken for the particles to travel this distance: Using the formula, time = distance/speed, we have:

   $t = \frac{45 \text{ m}}{0.97c} = \frac{45}{0.97 \times 3 \times 10^8} \approx 1.5 \times 10^{-7} \text{ s}$

   3. Half-life calculation: Given the intensity at the second detector is 12.5% of the first, we can relate this to the half-lives. Since each half-life reduces the intensity by half,

   $I = I_0 \left( \frac{1}{2} \right)^{n}$

   We solve for n:

   $0.125 = \left( \frac{1}{2} \right)^{n} \Rightarrow n = 3$ Therefore, the half-life relates to about 3 intervals. The time taken was approximately $1.5 \times 10^{-7}$ s, so:

   $\text{Half-life} = \frac{t}{3} = \frac{1.5 \times 10^{-7}}{3} \approx 5.0 \times 10^{-8} \text{ s}$

The proper time in this context is the time measured in the reference frame where the particles are at rest, which is the time taken for the particle beam to travel between the detectors in the lab frame.