5. The $$ \mathrm{K}+ $$ meson, a subatomic particle, has an average rest lifetime of $$ 1.0 imes 10^{-8} \mathrm{~s} $$. If the particle travels through the laboratory at $$ 2.6 imes 10^{8} \mathrm{~m} / \mathrm{s} $$, by how much has its lifetime, relative to the laboratory, increased?

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5. The $$ \mathrm{K}+ $$ meson, a subatomic particle, has an average rest lifetime of $$ 1.0 	imes 10^{-8} \mathrm{~s} $$. If the particle travels through the laboratory at $$ 2.6 	imes 10^{8} \mathrm{~m} / \mathrm{s} $$, by how much has its lifetime, relative to the laboratory, increased?

UpStudy AI · Accepted Answer

To determine how much the lifetime of the $$ \mathrm{K}^+ $$ meson has increased relative to the laboratory, we'll use the concept of **time dilation** from Einstein's theory of **special relativity**.

### Given:
- **Proper lifetime ($$ \tau_0 $$)**: $$ 1.0 \times 10^{-8} $$ seconds (lifetime in the particle's rest frame)
- **Velocity ($$ v $$)**: $$ 2.6 \times 10^8 $$ meters per second
- **Speed of light ($$ c $$)**: Approximately $$ 3.0 \times 10^8 $$ meters per second

### Time Dilation Formula:
The relationship between the proper lifetime ($$ \tau_0 $$) and the dilated lifetime ($$ \tau $$) in the laboratory frame is given by:
$$
\tau = \gamma \tau_0
$$
where $$ \gamma $$ (the Lorentz factor) is:
$$
\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}
$$

### Calculations:
1. **Calculate $$ \beta $$**:
$$
\beta = \frac{v}{c} = \frac{2.6 \times 10^8 \, \text{m/s}}{3.0 \times 10^8 \, \text{m/s}} \approx 0.8667
$$

2. **Calculate $$ \gamma $$**:
$$
\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - (0.8667)^2}} = \frac{1}{\sqrt{1 - 0.75}} = \frac{1}{\sqrt{0.25}} = 2
$$

3. **Determine the Dilated Lifetime ($$ \tau $$)**:
$$
\tau = \gamma \tau_0 = 2 \times 1.0 \times 10^{-8} \, \text{s} = 2.0 \times 10^{-8} \, \text{s}
$$

### Conclusion:
The lifetime of the $$ \mathrm{K}^+ $$ meson, as observed in the laboratory, is **twice** its proper lifetime due to the effects of time dilation.

**Answer:**  
Its lifetime is doubled in the laboratory frame; the laboratory-measured lifetime is twice τ₀.
The $$ \mathrm{K}^+ $$ meson's lifetime increases by a factor of 2 in the laboratory frame due to time dilation.

The meson, a subatomic particle, has an average rest lifetime of .
If the particle travels through the laboratory at , by how much has
its lifetime, relative to the laboratory, increased?

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The meson, a subatomic particle, has an average rest lifetime of . If the particle travels through the laboratory at , by how much has its lifetime, relative to the laboratory, increased?