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Chapter 26: Problem 32

Electron A is moving west with speed \(\frac{3}{5} c\) relative to the lab. Electron \(\mathrm{B}\) is also moving west with speed \(\frac{4}{5} c\) relative to the lab. What is the speed of electron \(\mathrm{B}\) in a frame of reference in which electron \(\mathrm{A}\) is at rest?

Short Answer

Expert verified
Answer: The speed of electron B in a frame of reference where electron A is at rest is \(\frac{5}{13}c\).

Step by step solution

01

Identify given information

Electron A is moving west with a speed of \(\frac{3}{5}c\) relative to the lab and electron B is moving west with a speed of \(\frac{4}{5}c\) relative to the lab. Our goal is to find the speed of electron B in a frame of reference where electron A is at rest.
02

Apply the relativistic velocity addition formula

The relativistic velocity addition formula states: $$ V_{B/A} = \frac{V_B - V_A}{1 - \frac{V_A V_B}{c^2}} $$ Where \(V_A\) is the velocity of electron A in the lab frame, \(V_B\) is the velocity of electron B in the lab frame, \(V_{B/A}\) is the relative velocity of electron B with respect to electron A, and \(c\) is the speed of light.
03

Plug in the given values

Now, we will plug in the given values for electron A and electron B. Since they are both moving west, we can use their given speeds directly in the formula without worrying about their signs: $$ V_{B/A} = \frac{\frac{4}{5}c - \frac{3}{5}c}{1 - \frac{(\frac{3}{5}c)(\frac{4}{5}c)}{c^2}} $$
04

Solve for the relative velocity

Simplifying the equation, we get: $$ V_{B/A} = \frac{\frac{1}{5}c}{1 - \frac{12}{25}} $$ And finally, $$ V_{B/A} = \frac{\frac{1}{5}c}{\frac{13}{25}} = \frac{5}{13}c $$ Therefore, the speed of electron B in a frame of reference in which electron A is at rest is \(\frac{5}{13}c\).

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