Question
A spaceship is traveling at 0.600c relative to the Earth. A passenger on the spaceship measures the length of the spaceship to be 120 meters. Calculate the length of the spaceship as observed by an Earth-based observer.
Answers
GPT-4o mini
To calculate the length of the spaceship as observed by an Earth-based observer, we need to use the concept of length contraction from the theory of special relativity. According to this theory, the length of an object measured in a frame where the object is moving is shorter than the length measured in the object's own rest frame.
The formula for length contraction is given by:
\[
L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
\]
where:
- \( L \) is the contracted length (length measured by the observer on Earth).
- \( L_0 \) is the proper length (length measured by the passenger on the spaceship).
- \( v \) is the velocity of the spaceship relative to the observer (Earth).
- \( c \) is the speed of light.
Given the values:
- \( L_0 = 120 \) meters (proper length measured by the passenger).
- \( v = 0.600c \).
First, we need to calculate \( v^2/c^2 \):
\[
\frac{v^2}{c^2} = (0.600)^2 = 0.36
\]
Now we find \( 1 - \frac{v^2}{c^2} \):
\[
1 - \frac{v^2}{c^2} = 1 - 0.36 = 0.64
\]
Now take the square root:
\[
\sqrt{1 - \frac{v^2}{c^2}} = \sqrt{0.64} = 0.8
\]
Now we can substitute \( L_0 \) and the square root into the length contraction formula:
\[
L = L_0 \sqrt{1 - \frac{v^2}{c^2}} = 120 \, \text{m} \cdot 0.8 = 96 \, \text{m}
\]
Thus, the length of the spaceship as observed by an Earth-based observer is:
\[
\boxed{96 \, \text{meters}}
\]
The formula for length contraction is given by:
\[
L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
\]
where:
- \( L \) is the contracted length (length measured by the observer on Earth).
- \( L_0 \) is the proper length (length measured by the passenger on the spaceship).
- \( v \) is the velocity of the spaceship relative to the observer (Earth).
- \( c \) is the speed of light.
Given the values:
- \( L_0 = 120 \) meters (proper length measured by the passenger).
- \( v = 0.600c \).
First, we need to calculate \( v^2/c^2 \):
\[
\frac{v^2}{c^2} = (0.600)^2 = 0.36
\]
Now we find \( 1 - \frac{v^2}{c^2} \):
\[
1 - \frac{v^2}{c^2} = 1 - 0.36 = 0.64
\]
Now take the square root:
\[
\sqrt{1 - \frac{v^2}{c^2}} = \sqrt{0.64} = 0.8
\]
Now we can substitute \( L_0 \) and the square root into the length contraction formula:
\[
L = L_0 \sqrt{1 - \frac{v^2}{c^2}} = 120 \, \text{m} \cdot 0.8 = 96 \, \text{m}
\]
Thus, the length of the spaceship as observed by an Earth-based observer is:
\[
\boxed{96 \, \text{meters}}
\]