A cosmic ray muon travels down through the atmosphere,travelling from the top of a 5.0-km tall mountain to sea level in 17 $\mu$s as measured by astronomers on Earth.What height would one travelling with the muon observe the mountain to be? Use 3.0 * 10⁸ for the speed of light.

Question

Quizplus · Accepted Answer

To find the observed height of the mountain from the muon's perspective, we use the concept of length contraction in special relativity. The formula for length contraction is $L = L_0 \sqrt{1 - \frac{v^2}{c^2}}$, where $L$ is the contracted length (the height of the mountain as observed by the muon), $L_0$ is the proper length (the actual height of the mountain, 5.0 km), $v$ is the velocity of the muon, and $c$ is the speed of light. Given the time it takes for the muon to travel the height of the mountain as observed on Earth is $1.7 \times 10^{-5}$ seconds, and the speed of light is $3.0 \times 10^8$ m/s, we first find the velocity of the muon as $v = \frac{distance}{time} = \frac{5000 m}{1.7 \times 10^{-5} s}$. Then, we substitute $v$ and $c$ into the length contraction formula to find $L$. The calculation shows that the observed height is significantly contracted, leading to the correct answer being 0.99 km, which accounts for the relativistic effects of length contraction at high velocities.

A Cosmic Ray Muon Travels Down Through the Atmosphere,travelling from the Top