A meson when at rest decays 2 `mus` after it is created. If moving in the laboratory at 0.99c, its lifetime according to laboratory clocks would be

Options

A. the same
B. 0.28 s
C. 14 `mus`
D. 4.6 s

Answer

To solve the problem of determining the lifetime of a meson moving at a speed of 0.99c, we will use the concept of time dilation from the theory of relativity.

### Step-by-Step Solution:

1. **Understand the Rest Lifetime of the Meson**:
The meson has a rest lifetime (proper time) of \( \Delta t_0 = 2 \, \mu s \) (microseconds). This is the time it takes for the meson to decay when it is at rest.

2. **Identify the Speed of the Meson**:
The meson is moving at a speed of \( v = 0.99c \), where \( c \) is the speed of light.

3. **Use the Time Dilation Formula**:
The time dilation formula relates the proper time \( \Delta t_0 \) to the dilated time \( \Delta t \) observed in a different frame (the laboratory frame in this case):
\[ \Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

4. **Calculate \( \sqrt{1 - \frac{v^2}{c^2}} \)**:
First, calculate \( \frac{v^2}{c^2} \):
\[ \frac{v^2}{c^2} = (0.99)^2 = 0.9801 \]
Now, calculate \( 1 - \frac{v^2}{c^2} \):
\[ 1 - 0.9801 = 0.0199 \]
Now, take the square root:
\[ \sqrt{0.0199} \approx 0.141 \]

5. **Calculate the Dilated Time**:
Substitute the values into the time dilation formula:
\[ \Delta t = \frac{2 \, \mu s}{0.141} \approx 14.18 \, \mu s \]

6. **Final Answer**:
The lifetime of the meson according to laboratory clocks is approximately \( 14.18 \, \mu s \).