A spaceship traveling at 0.53c relative to the Earth sends out a rocket that travels at 0.75 as seen by viewers on Earth. If the rocket is traveling the same direction as the ship, how fast does it appear to be traveling according to viewers on the ship? give the answer to 3 decimal places, in terms of c:____________C
A spaceship traveling at 0.53c relative to the Earth sends out a rocket that travels at 0.75 as seen by viewers on Earth. If the rocket is traveling the same direction as the ship, how fast does it appear to be traveling according to viewers on the ship? give the answer to 3 decimal places, in terms of c:____________C
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![### Problem Statement
A spaceship traveling at \(0.53c\) relative to the Earth sends out a rocket that travels at \(0.75c\) as seen by viewers on Earth. If the rocket is traveling in the same direction as the ship, how fast does it appear to be traveling according to viewers on the ship? Give the answer to three decimal places, in terms of \(c\): \( \_\_\_\_\_ c \).
### Explanation
This problem involves the concept of relativistic velocity addition in the context of special relativity. The key formula for adding velocities \(u\) and \(v\) in the same direction when dealing with relativistic speeds (close to the speed of light, \(c\)) is:
\[ u' = \frac{u + v}{1 + \frac{uv}{c^2}} \]
where:
- \(u'\) is the velocity of the rocket as observed from the spaceship.
- \(u\) is the velocity of the rocket as observed from the Earth (\(0.75c\)).
- \(v\) is the velocity of the spaceship relative to the Earth (\(0.53c\)).
### Calculation
Plugging in the values:
\[ u' = \frac{0.75c + 0.53c}{1 + \frac{(0.75c)(0.53c)}{c^2}} \]
Simplifying inside the fraction:
\[ u' = \frac{0.75c + 0.53c}{1 + 0.3975} \]
\[ u' = \frac{1.28c}{1.3975} \]
Further simplifying:
\[ u' \approx 0.916c \]
### Answer
The apparent speed of the rocket as observed from the spaceship is \(0.916c\).
### Key Concepts
- **Relativistic Velocity Addition**: When velocities approach the speed of light, they do not simply add up as in classical mechanics. Instead, the formula above must be used to account for the effects of special relativity.
- **Speed of Light**: \(c\), approximately \(3 \times 10^8\) meters per second, is the maximum speed limit in the universe and plays a crucial role in relativistic equations.
### Application
This calculation is important in understanding how speeds are perceived differently by observers in different reference frames, which](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed7c40be-308c-4497-87ab-a20f4a14a0a8%2F3713862c-edba-4ce8-a3c3-0671c5c44a31%2Ffl1imlm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A spaceship traveling at \(0.53c\) relative to the Earth sends out a rocket that travels at \(0.75c\) as seen by viewers on Earth. If the rocket is traveling in the same direction as the ship, how fast does it appear to be traveling according to viewers on the ship? Give the answer to three decimal places, in terms of \(c\): \( \_\_\_\_\_ c \).
### Explanation
This problem involves the concept of relativistic velocity addition in the context of special relativity. The key formula for adding velocities \(u\) and \(v\) in the same direction when dealing with relativistic speeds (close to the speed of light, \(c\)) is:
\[ u' = \frac{u + v}{1 + \frac{uv}{c^2}} \]
where:
- \(u'\) is the velocity of the rocket as observed from the spaceship.
- \(u\) is the velocity of the rocket as observed from the Earth (\(0.75c\)).
- \(v\) is the velocity of the spaceship relative to the Earth (\(0.53c\)).
### Calculation
Plugging in the values:
\[ u' = \frac{0.75c + 0.53c}{1 + \frac{(0.75c)(0.53c)}{c^2}} \]
Simplifying inside the fraction:
\[ u' = \frac{0.75c + 0.53c}{1 + 0.3975} \]
\[ u' = \frac{1.28c}{1.3975} \]
Further simplifying:
\[ u' \approx 0.916c \]
### Answer
The apparent speed of the rocket as observed from the spaceship is \(0.916c\).
### Key Concepts
- **Relativistic Velocity Addition**: When velocities approach the speed of light, they do not simply add up as in classical mechanics. Instead, the formula above must be used to account for the effects of special relativity.
- **Speed of Light**: \(c\), approximately \(3 \times 10^8\) meters per second, is the maximum speed limit in the universe and plays a crucial role in relativistic equations.
### Application
This calculation is important in understanding how speeds are perceived differently by observers in different reference frames, which
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