### Understanding Relative Velocity in Opposite Directions#### Problem Statement:Spaceships A and B move in opposite directions at the same speed of \( 0.378c \) relative to Earth, with A moving away from Earth and B moving towards Earth. Find the velocity of B relative to A.#### Solution:To solve this problem, we can use the concept of relative velocity from the theory of special relativity. In special relativity, the addition of velocities is given by the formula:\[ v_{rel} = \frac{v + u}{1 + \frac{vu}{c^2}} \]Where:- \( v_{rel} \) is the relative velocity of one object with respect to the other.- \( v \) and \( u \) are the velocities of the two objects relative to the Earth.- \( c \) is the speed of light.Given:- \( v = 0.378c \) (velocity of spaceship A relative to Earth)- \( u = -0.378c \) (velocity of spaceship B relative to Earth, negative because it moves in the opposite direction)By substituting these values into the formula, we get:\[ v_{rel} = \frac{0.378c + (-0.378c)}{1 + \frac{(0.378c)(-0.378c)}{c^2}} \]Since the velocities are equal but opposite:\[ v_{rel} = \frac{0.378c + 0.378c}{1 - \left(\frac{0.378^2}{c^2}\right)} \]Simplifying further:\[ v_{rel} = \frac{0.756c}{1 - 0.142884} \]\[ v_{rel} = \frac{0.756c}{0.857116} \]Finally:\[ v_{rel} \approx 0.882c \]So, the velocity of spaceship B relative to spaceship A is approximately \( 0.882c \).#### Diagram:*No diagram is provided, but the problem can be visualized with two spaceships moving in opposite directions and Earth as a reference point at the center.*### Conclusion:By using the relativistic velocity addition formula, we have determined that the velocity of spaceship B relative to spaceship A is approximately \( 0.882c \). This example demonstrates how special relativity

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Spaceships A and B move in opposite directions at the same speed of 0.378c relative to earth, with A moving away from earth and B moving toward earth. Find the velocity of B relative to A. с

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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