Understanding Length Contraction
* Proper Length (L₀): The length of an object measured by an observer at rest relative to the object.
* Relativistic Length (L): The length of an object measured by an observer moving at a relative velocity (v) with respect to the object.
* Length Contraction Formula: L = L₀√(1 - (v²/c²)), where:
* L is the relativistic length
* L₀ is the proper length
* v is the relative velocity
* c is the speed of light
Solving the Problem
1. Given:
* L = L₀/2 (The relativistic length is half the proper length)
2. Substitute into the formula:
L₀/2 = L₀√(1 - (v²/c²))
3. Simplify:
1/2 = √(1 - (v²/c²))
4. Square both sides:
1/4 = 1 - (v²/c²)
5. Rearrange to solve for v²/c²:
(v²/c²) = 3/4
6. Take the square root of both sides:
v/c = √(3/4) = √3/2
7. Solve for v:
v = (√3/2) * c
Therefore, the speed of the observer relative to the rod is (√3/2) times the speed of light.
