A cosmic ray particle travels directly over a football field, from one goal line to the other, at a speed of \(0.50 c .\) (a) If the length of the field between goal lines in the Earth frame is \(91.5 \mathrm{m}(100 \mathrm{yd}),\) what length is measured in the rest frame of the particle? (b) How long does it take the particle to go from one goal line to the other according to Earth observers? (c) How long does it take in the rest frame of the particle?

(a) Calculate the contracted length of the football field in the particle's rest frame. To find the contracted length, we first calculate the Lorentz factor: \(\gamma = \frac{1}{\sqrt{1 - \frac{(0.50 c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.5^2}} = \frac{1}{\sqrt{0.75}} \approx 1.155\) Applying the length contraction formula, we get: \(L' = \frac{91.5 \, \mathrm{m}}{\gamma} = \frac{91.5 \, \mathrm{m}}{1.155} \approx 79.2 \, \mathrm{m}\) The length of the football field measured in the rest frame of the particle is approximately 79.2 meters. (b) Calculate the time taken for the particle to travel from one goal line to the other in the Earth frame. Using the given speed and length, we calculate the time taken in the Earth frame: \(t = \frac{91.5 \, \mathrm{m}}{0.50 c} = \frac{91.5 \, \mathrm{m}}{0.50 \times 3 \times 10^8 \, \mathrm{m/s}} \approx 6.1 \times 10^{-7} \, \mathrm{s}\) The particle takes approximately \(6.1 \times 10^{-7}\) seconds to travel the length of the football field in the Earth frame. (c) Calculate the time taken for the particle to travel from one goal line to the other in the particle's rest frame. Using the time dilation formula, we calculate the time taken in the particle's rest frame: \(t' = \frac{\frac{91.5 \, \mathrm{m}}{0.50 c}}{\gamma} = \frac{6.1 \times 10^{-7} \, \mathrm{s}}{1.155} \approx 5.3 \times 10^{-7} \, \mathrm{s}\) The time taken for the particle to travel the length of the football field in its rest frame is approximately \(5.3 \times 10^{-7}\) seconds.