Suppose an astronaut, such as the twin in the twin paradox discussion, travels so fast that y=30.00. The astronaut travels from Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an earthbound observer. How far apart are Earth and Alpha Centauri as measured by the astronaut?
Suppose an astronaut, such as the twin in the twin paradox discussion, travels so fast that y=30.00. The astronaut travels from Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an earthbound observer. How far apart are Earth and Alpha Centauri as measured by the astronaut?
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![**Scenario Overview:**
The scenario describes an astronaut traveling at a high velocity, similar to the twin in the "twin paradox" concept in relativity. The astronaut's velocity results in a Lorentz factor (\(\gamma\)) of 30. This factor is used in relativity to account for time dilation and length contraction effects experienced at relativistic speeds. The astronaut's journey is from Earth to Alpha Centauri, a star system located 4.300 light years away, according to an earthbound observer.
**Problem Statement:**
The question asks for the distance between Earth and Alpha Centauri as measured by the astronaut, taking into account the effects of special relativity.
**Key Concept: Length Contraction**
In special relativity, the observed length of an object in motion is shorter than its rest length. This is given by the formula:
\[ L = \frac{L_0}{\gamma} \]
where:
- \( L \) is the contracted length observed by the astronaut.
- \( L_0 \) is the proper length measured by a stationary observer (4.300 light years in this case).
- \( \gamma \) is the Lorentz factor (30 in this scenario).
Thus, to find the distance as measured by the astronaut:
\[ L = \frac{4.300 \, \text{ly}}{30} \]
By applying this formula, you can calculate how far the astronaut perceives the journey to be.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F440b7269-170b-4024-8d0e-4b8a2f594bf7%2Fb1ea0ccd-d0d3-494e-8abf-c0b3fbeb9ed9%2F7sygyc_processed.png&w=3840&q=75)
Transcribed Image Text:**Scenario Overview:**
The scenario describes an astronaut traveling at a high velocity, similar to the twin in the "twin paradox" concept in relativity. The astronaut's velocity results in a Lorentz factor (\(\gamma\)) of 30. This factor is used in relativity to account for time dilation and length contraction effects experienced at relativistic speeds. The astronaut's journey is from Earth to Alpha Centauri, a star system located 4.300 light years away, according to an earthbound observer.
**Problem Statement:**
The question asks for the distance between Earth and Alpha Centauri as measured by the astronaut, taking into account the effects of special relativity.
**Key Concept: Length Contraction**
In special relativity, the observed length of an object in motion is shorter than its rest length. This is given by the formula:
\[ L = \frac{L_0}{\gamma} \]
where:
- \( L \) is the contracted length observed by the astronaut.
- \( L_0 \) is the proper length measured by a stationary observer (4.300 light years in this case).
- \( \gamma \) is the Lorentz factor (30 in this scenario).
Thus, to find the distance as measured by the astronaut:
\[ L = \frac{4.300 \, \text{ly}}{30} \]
By applying this formula, you can calculate how far the astronaut perceives the journey to be.
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