Textbook Question
(II) What is the speed of an electron after being accelerated from rest by 28,000 V?
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35. Special Relativity
Inertial Reference Frames
Problem 46b
Textbook Question
The star Alpha goes supernova. Ten years later and 100 ly away, as measured by astronomers in the galaxy, star Beta explodes. An alien spacecraft passing through the galaxy finds that the distance between the two explosions is 120 ly. According to the aliens, what is the time between the explosions?
Verified step by step guidance1
Step 1: Recognize that this problem involves special relativity, specifically the concept of spacetime intervals. The spacetime interval (Δs²) is invariant between reference frames and is given by the equation: , where Δx is the spatial separation, Δt is the time separation, and c is the speed of light.
Step 2: In the galaxy's reference frame, the spatial separation between the two explosions is 100 light-years (ly), and the time separation is 10 years. Calculate the spacetime interval in the galaxy's frame using: . Note that c = 1 ly/year in natural units.
Step 3: In the alien spacecraft's reference frame, the spatial separation is given as 120 ly. Since the spacetime interval is invariant, use the same equation: , where Δt' is the time separation in the alien's frame. Set this equal to the spacetime interval calculated in Step 2.
Step 4: Solve for Δt' in the alien's frame. Rearrange the equation: . Take the square root to find: . Substitute the value of Δs² from Step 2.
Step 5: Simplify the expression to find the time separation Δt' in the alien's frame. Ensure all units are consistent (e.g., using natural units where c = 1 ly/year). This will give the time between the explosions as measured by the aliens.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Light Year
A light year is a unit of distance that represents how far light travels in one year, approximately 5.88 trillion miles (9.46 trillion kilometers). In the context of astronomy, it is used to measure vast distances between celestial objects. Understanding light years is crucial for interpreting the distances mentioned in the question, as it helps to relate the time taken for light to travel from one event to another.
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Supernova
A supernova is a powerful and luminous explosion that occurs at the end of a star's life cycle, often resulting in the star's destruction. This event releases an enormous amount of energy and can outshine entire galaxies for a short period. Recognizing the significance of supernovae is essential for understanding the timeline of events in the question, as it marks the starting point for measuring the time until the next explosion.
Relativity of Simultaneity
The relativity of simultaneity is a concept from Einstein's theory of relativity, which states that events that are simultaneous in one frame of reference may not be simultaneous in another. This principle is important in the context of the question, as the alien spacecraft measures the time between the two explosions differently than observers on Earth, due to the finite speed of light and the distances involved.
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