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Chapter 35: Problem 28

If a muon is moving at \(90.0 \%\) of the speed of light, how does its measured lifetime compare to when it is in the rest frame of a laboratory, where its lifetime is \(2.2 \cdot 10^{-6}\) s?

Short Answer

Expert verified
Answer: When moving at 90% the speed of light, the measured lifetime of a muon is approximately 5.046 microseconds, which is more than 2 times longer than its lifetime in the rest frame (2.2 microseconds).

Step by step solution

01

Identify the time dilation formula

The time dilation formula, as per special relativity, is given by: \(T = T_{0} \cdot \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\), where T is the dilated time (measured time), \(T_{0}\) is the proper time (time in the rest frame), v is the relative velocity of the moving frame, and c is the speed of light.
02

Determine the moving frame's relative velocity

The muon is moving at 90% of the speed of light. To find its relative velocity v, we multiply the speed of light (c) by 0.9 (90% in decimal form): \(v = 0.9c\).
03

Substitute values in the time dilation formula

We have the proper time \(T_{0} = 2.2 \cdot 10^{-6}\) s, and the relative velocity \(v = 0.9c\). Substitute these values into the equation: \(T = (2.2 \cdot 10^{-6}) \cdot \frac{1}{\sqrt{1-\frac{(0.9c)^2}{c^2}}}\).
04

Simplify the equation

Now we need to simplify the expression in the equation: \(T = (2.2 \cdot 10^{-6}) \cdot \frac{1}{\sqrt{1-\frac{(0.81c^2)}{c^2}}}\).
05

Cancel the terms and calculate the time dilation factor

The \(c^2\) in the numerator and the denominator cancel each other out: \(T = (2.2 \cdot 10^{-6}) \cdot \frac{1}{\sqrt{1-0.81}}\). Calculate the time dilation factor: \(\frac{1}{\sqrt{1-0.81}} = \frac{1}{\sqrt{0.19}} \approx 2.294\).
06

Determine the measured lifetime of the moving muon

Multiply the proper time by the time dilation factor: \(T \approx (2.2 \cdot 10^{-6}) \cdot 2.294 \approx 5.046 \cdot 10^{-6}\) s. So, the measured lifetime of the muon when it is moving at 90% the speed of light is approximately 5.046 microseconds.
07

Compare the measured lifetime to the lifetime in the rest frame

The lifetime in the rest frame of the laboratory is 2.2 microseconds, while the measured lifetime when it is moving at 90% of the speed of light is approximately 5.046 microseconds. This means that the muon's lifetime appears to be more than 2 times longer when moving at a high speed compared to its lifetime in the rest frame.

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Most popular questions from this chapter

Show that momentum and energy transform from one inertial frame to another as \(p_{x}^{\prime}=\gamma\left(p_{x}-v E / c^{2}\right) ; p_{y}^{\prime}=p_{y}\) \(p_{z}^{\prime}=p_{p} ; E^{\prime}=\gamma\left(E-v p_{x}\right) .\) Hint: Look at the derivation for the space-time Lorentz transformation.

In Jules Verne's classic Around the World in Eighty Days, Phileas Fogg travels around the world in, according to his calculation, 81 days. Due to crossing the International Date Line he actually made it only 80 days. How fast would he have to go in order to have time dilation make 80 days to seem like \(81 ?\) (Of course, at this speed, it would take a lot less than even 1 day to get around the world \(\ldots . .)\)

Robert, standing at the rear end of a railroad car of length \(100 . \mathrm{m},\) shoots an arrow toward the front end of the car. He measures the velocity of the arrow as \(0.300 c\). Jenny, who was standing on the platform, saw all of this as the train passed her with a velocity of \(0.750 c .\) Determine the following as observed by Jenny: a) the length of the car b) the velocity of the arrow c) the time taken by arrow to cover the length of the car d) the distance covered by the arrow

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