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Chapter 35: Problem 25

An astronaut on a spaceship traveling at a speed of \(0.50 c\) is holding a meter stick parallel to the direction of motion. a) What is the length of the meter stick as measured by another astronaut on the spaceship? b) If an observer on Earth could observe the meter stick, what would be the length of the meter stick as measured by that observer?

Short Answer

Expert verified
Answer: a) The length observed by another astronaut on the spaceship is 1 meter. b) The length of the meter stick as observed by an Earth-bound observer is approximately 0.866 meters.

Step by step solution

01

Part a: Length in the Astronaut's Rest Frame

In this case, the astronaut holding the meter stick and the observer (another astronaut) on the same spaceship are both in the same rest frame. Since they are moving together, there is no relative velocity between them. Hence, the length observed by the second astronaut will be the same as the proper length of the meter stick which is 1 meter. So, the length of the meter stick observed by the second astronaut is 1 meter.
02

Part b: Length Observed by an Earth-Bound Observer

Now, we need to find out the length of the meter stick as observed by an Earth-bound observer. In this case, the relative velocity between the observer on Earth and the meter stick is 0.50 c; therefore, we can use the length contraction formula given above to find the contracted length of the meter stick as seen by this observer. Plugging in the values into the formula: $$ L = L_0\sqrt{1-\frac{v^2}{c^2} } $$ $$ L = 1\sqrt{1-\frac{(0.50c)^2}{c^2} } $$ $$ L = \sqrt{1-\frac{0.25c^2}{c^2} } $$ $$ L = \sqrt{1-0.25} $$ $$ L = \sqrt{0.75} $$ $$ L \approx 0.866$$ So, the length of the meter stick as measured by the Earth-bound observer is approximately 0.866 meters.

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