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Chapter 26: Problem 6

An unstable particle called the pion has a mean lifetime of 25 ns in its own rest frame. A beam of pions travels through the laboratory at a speed of $0.60 c .$ (a) What is the mean lifetime of the pions as measured in the laboratory frame? (b) How far does a pion travel (as measured by laboratory observers) during this time?

Short Answer

Expert verified
Answer: The mean lifetime of a pion in the laboratory frame is 31.25 ns, and it travels approximately 5,625 meters during this time.

Step by step solution

01

Find the mean lifetime in the laboratory frame

We are given the mean lifetime of a pion in its rest frame, \(t_0 = 25\textrm{ ns}\) and its speed, \(v = 0.60c\). To find the mean lifetime in the laboratory frame, we will use the time dilation formula: \(t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}}\) where \(t\) is the time in the laboratory frame, \(t_0\) is the time in the pion's rest frame, \(v\) is the speed of the pion, and \(c\) is the speed of light. Plugging in the given values: \(t = \frac{25\textrm{ ns}}{\sqrt{1 - \frac{(0.60c)^2}{c^2}}}\)
02

Simplify the expression

Now, let's simplify the expression: \(t = \frac{25\textrm{ ns}}{\sqrt{1 - 0.36}}\) \(t = \frac{25\textrm{ ns}}{\sqrt{0.64}}\) \(t = \frac{25\textrm{ ns}}{0.8}\) \(t = 31.25\textrm{ ns}\) So, the mean lifetime of the pions as measured in the laboratory frame is \(31.25\textrm{ ns}\).
03

Calculate the distance traveled by the pions

Now that we have the mean lifetime of pions in the laboratory frame, we can calculate the distance they travel during this time. We use the formula: \(d = vt\) where \(d\) is the distance traveled, \(v\) is the speed of the pions, and \(t\) is the mean lifetime in the laboratory frame. Using the given speed and the calculated mean lifetime: \(d = (0.60c)(31.25\textrm{ ns})\)
04

Convert the time to distance using the speed of light

The speed of light, \(c\), is equal to approximately \(3.00\times10^8\textrm{ m/s}\), and there are \(10^9\textrm{ ns}\) in one second. Therefore, we can convert the time to distance using the speed of light: \(d = (0.60)(3.00\times10^8\textrm{ m/s})(31.25\textrm{ ns})\) \(d = (0.60)(3.00\times10^8\textrm{ m/s})(31.25\times10^{-9}\textrm{ s})\)
05

Calculate the distance

Finally, let's calculate the distance: \(d = (0.60)(3.00\times10^8\textrm{ m})(31.25\times10^{-9})\) \(d \approx 5.625\times10^3\textrm{ m}\) So, a pion travels approximately \(5.625\times10^3\textrm{ m}\), as measured by laboratory observers during its mean lifetime of \(31.25\textrm{ ns}\) in the laboratory frame.

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