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

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.