After being produced, a neutral pion must travel down a tube that is 100 m long to the experimental area. The pion has a lifetime T= 2 x 10 sec in its rest frame. How fast must the pion travel if it is not to decay before it reaches the end of the tube? In other words, what is the minimum speed vin that the pion must have if it is to reach the end of the tube before decaying? Give your answer to 3 significant figures as a multiple of c, the speed of light.
After being produced, a neutral pion must travel down a tube that is 100 m long to the experimental area. The pion has a lifetime T= 2 x 10 sec in its rest frame. How fast must the pion travel if it is not to decay before it reaches the end of the tube? In other words, what is the minimum speed vin that the pion must have if it is to reach the end of the tube before decaying? Give your answer to 3 significant figures as a multiple of c, the speed of light.
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### Detailed Explanation
To find the minimum speed \(v_{\text{min}}\) of the pion that ensures it will travel the full 100 m length of the tube before decaying, consider both the lifetime of the pion in its rest frame and the distance it must cover within this frame.
#### Variables and Given Data:
- Distance to travel, \(L = 100\) m
- Lifetime in rest frame, \(\tau = 2 \times 10^{-8}\) sec
#### Steps to solve:
1. **Relativistic Time Dilation:**
- In the laboratory frame, the pion's lifetime \(\tau_{\text{lab}}\) is longer due to time dilation:
\[
\tau_{\text{lab}} = \gamma \tau
\]
where \(\gamma\) is the Lorentz factor:
\[
\gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}
\]
2. **Distance and Speed Relationship:**
- The pion must travel distance \(L\) within its dilated lifetime \(\tau_{\text{lab}}\):
\[
v = \frac{L}{\tau_{\text{lab}}}
\]
3. **Combining equations:**
- Substitute \(\tau_{\text{lab}}\):
\[
\tau_{\text{lab}} = \frac{L}{v}
\]
hence,
\[
\frac{L}{v} = \gamma \tau
\]
- Substitute \(\gamma\):
\[
\frac{L}{v} = \frac{\tau}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aaf72ae-8073-4cb4-ae75-7d87fdc3f506%2F71828f8a-9442-4512-9a5d-83e2281f1e38%2F9b9vdq4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
After being produced, a neutral pion must travel down a tube that is 100 m long to the experimental area. The pion has a lifetime \(\tau = 2 \times 10^{-8}\) sec in its rest frame. How fast must the pion travel if it is not to decay before it reaches the end of the tube? In other words, what is the minimum speed \(v_{\text{min}}\) that the pion must have if it is to reach the end of the tube before decaying? Give your answer to 3 significant figures as a multiple of \(c\), the speed of light.
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### Detailed Explanation
To find the minimum speed \(v_{\text{min}}\) of the pion that ensures it will travel the full 100 m length of the tube before decaying, consider both the lifetime of the pion in its rest frame and the distance it must cover within this frame.
#### Variables and Given Data:
- Distance to travel, \(L = 100\) m
- Lifetime in rest frame, \(\tau = 2 \times 10^{-8}\) sec
#### Steps to solve:
1. **Relativistic Time Dilation:**
- In the laboratory frame, the pion's lifetime \(\tau_{\text{lab}}\) is longer due to time dilation:
\[
\tau_{\text{lab}} = \gamma \tau
\]
where \(\gamma\) is the Lorentz factor:
\[
\gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}
\]
2. **Distance and Speed Relationship:**
- The pion must travel distance \(L\) within its dilated lifetime \(\tau_{\text{lab}}\):
\[
v = \frac{L}{\tau_{\text{lab}}}
\]
3. **Combining equations:**
- Substitute \(\tau_{\text{lab}}\):
\[
\tau_{\text{lab}} = \frac{L}{v}
\]
hence,
\[
\frac{L}{v} = \gamma \tau
\]
- Substitute \(\gamma\):
\[
\frac{L}{v} = \frac{\tau}{
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