Question

Refer to Example \(26.2 .\) One million muons are moving toward the ground at speed \(0.9950 c\) from an altitude of \(4500 \mathrm{m} .\) In the frame of reference of an observer on the ground, what are (a) the distance traveled by the muons; (b) the time of flight of the muons; (c) the time interval during which half of the muons decay; and (d) the number of muons that survive to reach sea level. [Hint: The answers to (a) to (c) are not the same as the corresponding quantities in the muons' reference frame. Is the answer to (d) the same?]

Short answer

Answer: The time interval during which half of the muons decay in the reference frame of an observer on the ground can be found in step 4, where we calculate the time dilation of the decay time using the Lorentz factor and the decay time in the muons' reference frame. The time interval is represented as ∆t = γ × τ, where τ represents the known decay time in the muons' reference frame (2.2 × 10^(-6) s), and γ represents the Lorentz factor found in step 1.

Step-by-step solution

Calculate the Lorentz factor

Given the speed of the muons as \(0.9950c\), first we need to calculate the Lorentz factor, which quantifies the time dilation and length contraction. The Lorentz factor \(\gamma\) can be calculated by the following formula: \(\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\) With \(v = 0.9950c\), we can find \(\gamma\).

Calculate the distance traveled in the reference frame of an observer on the ground

The distance traveled by the muons in the reference frame of an observer on the ground should be the same as the altitude they drop from, which is \(4500m\).

Calculate the time of flight in the reference frame of an observer on the ground

Using the equation \(d=v\cdot t\), we can find the time of flight in the reference frame of an observer on the ground: \(t=\frac{d}{v}\) With \(d=4500\mathrm{m}\) and \(v=0.9950c\), we can find the time of flight \(t\).

Calculate the proper time interval for half of the muons to decay

In the muons' reference frame, we know that their decay time is \(\tau=2.2\cdot10^{-6}\textrm{s}\). From the Lorentz factor, we can find the time dilation of the decay time in the reference frame from an observer on the ground: \(\Delta t=\gamma\cdot\tau\) Using the calculated \(\gamma\), we can find the time interval \(\Delta t\) during which half of the muons decay in the reference frame of an observer on the ground.

Calculate the number of muons that survive to reach sea level

With the information from step 4, we can now calculate the number of muons that survive to reach sea level. Since half of the muons decay in time interval \(\Delta t\), we can find the number of muons decayed during the time of flight \(t\) from step 3: \(N_\textrm{decayed}=\frac{10^6}{2}\left(1-e^{-\frac{t}{\Delta t}}\right)\) And the surviving muons can be found by subtracting the decayed muons from the initial quantity: \(N_\textrm{survived}= 10^6 - N_\textrm{decayed}\)