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

To determine the half-life, we first need to calculate the time taken for the particles to travel between the two detectors.

1. Distance Between Detectors: The distance is given as 45 m.
2. Time Calculation: Using the formula for time, we have: $t = \frac{d}{v} = \frac{45 \, \text{m}}{0.97c}$ Using the speed of light, $c \approx 3 \times 10^8 \, \text{m/s}$, we find: $t = \frac{45}{0.97 \times 3 \times 10^8} \approx 1.54 \times 10^{-7} \, \text{s}$
3. Intensity Ratio and Half-Life relation: The intensity at the second detector is 12.5% of the first, which corresponds to: $I_2 = 0.125 I_1$ Given the intensity decay is linked to the half-life and recognizing that an intensity decrease to $0.125$ indicates the passage of 3 half-lives, we solve: $\text{Half-life} = \frac{t}{3} = \frac{1.54 \times 10^{-7}}{3} \approx 5.13 \times 10^{-8} \, \text{s}$.

The proper time in the context of this calculation is the time measured in the reference frame of the unstable particles, specifically when they are at rest relative to the detectors. This is observed as the time taken for the particle beam to travel between detectors.