Question

A person in a spaceship holds a meter stick parallel to the motion of the ship as it passes by the Earth with \(\gamma=2 .\) What length does an observer at rest on the Earth measure for the meter stick? a) \(2 \mathrm{~m}\) c) \(0.5 \mathrm{~m}\) e) none of the above b) \(1 \mathrm{~m}\) d) \(0.0 \mathrm{~m}\)

Short answer

Answer: 0.5 meters

Step-by-step solution

Recall the formula for length contraction

Length contraction occurs when an object is moving relative to an observer's frame of reference. The formula for length contraction is as follows: \[ L = L_{0} \cdot \sqrt{1 - v^2/c^2} \] where \(L_0\) is the proper length (the length of the object in its rest frame), \(L\) is the contracted length (the length observed by the moving observer), \(v\) is the relative velocity of the object, and \(c\) is the speed of light. Since we are given the value of \(\gamma\), we can also express this formula in terms of \(\gamma\): \[ L = \frac{L_{0}}{\gamma} \]

Identify the proper length

In this problem, the proper length (\(L_0\)) is the length of the meter stick as measured by the person in the spaceship. Since it is a meter stick, its proper length is \(L_0 = 1 \, \mathrm{m}\).

Calculate the contracted length

Substitute the values of \(L_0\) and \(\gamma\) into the formula for length contraction: \[ L = \frac{1 \, \mathrm{m}}{2} \]

Solve for L

Solve for the contracted length: \[ L = 0.5 \, \mathrm{m} \] The observer at rest on Earth will measure the length of the meter stick to be \(0.5 \, \mathrm{m}\). Therefore, the correct answer is (c) \(0.5 \, \mathrm{m}\).