To measure the length of the meter stick passing by at a speed of [tex]\(0.5c\)[/tex] (where [tex]\(c\)[/tex] is the speed of light), we need to consider the effects of special relativity, particularly the concept of Lorentz contraction.
### Step-by-Step Solution:
1. Understand the Rest Length:
- The rest length ([tex]\(L_0\)[/tex]) of the meter stick is 1 meter. This is the length measured when the stick is at rest relative to the observer.
2. Identify the Speed:
- The speed of the meter stick relative to the observer is [tex]\(0.5c\)[/tex].
3. Use the Lorentz Contraction Formula:
- According to the theory of relativity, objects moving at significant fractions of the speed of light will appear contracted in the direction of motion. The formula for Lorentz contraction is:
[tex]\[
L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
\][/tex]
where [tex]\(L\)[/tex] is the contracted length, [tex]\(L_0\)[/tex] is the rest length, [tex]\(v\)[/tex] is the velocity of the object, and [tex]\(c\)[/tex] is the speed of light.
4. Substitute the Known Values:
- Here, [tex]\(L_0 = 1.0\)[/tex] meter and [tex]\(v = 0.5c\)[/tex]:
[tex]\[
L = 1.0 \sqrt{1 - \left(0.5\right)^2}
\][/tex]
5. Compute the Term inside the Square Root:
- First, compute [tex]\((0.5)^2\)[/tex]:
[tex]\[
(0.5)^2 = 0.25
\][/tex]
- Then subtract this value from 1:
[tex]\[
1 - 0.25 = 0.75
\][/tex]
6. Calculate the Square Root:
- Take the square root of 0.75:
[tex]\[
\sqrt{0.75} \approx 0.8660254037844386
\][/tex]
7. Determine the Contracted Length:
- Multiply the rest length by the square root result:
[tex]\[
L = 1.0 \times 0.8660254037844386 \approx 0.8660254037844386 \text{ meters}
\][/tex]
### Conclusion:
When the meter stick passes by at a speed of [tex]\(0.5c\)[/tex], its length as measured by you will be approximately 0.866 meters. This contraction is a direct consequence of the relativistic effect encoded in Einstein's theory of special relativity.
