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Chapter 33: Problem 37

Use relativistic velocity addition to show that if an object moves at speed \(v

Short Answer

Expert verified
The relative speed of an object relative to any inertial frame is less than \(c\) if its speed is less than \(c\) in some other inertial frame. This is demonstrated using the formula for relativistic velocity addition, and it is a principle of special relativity.

Step by step solution

01

Understanding the Formula for Relativistic Velocity Addition

The formula for relativistic velocity addition is given by: \( u' = \frac{u + v}{1 + \frac{uv}{c^2}} \), where \(u'\) is the velocity of an object in a reference frame which is moving at speed \(v\) relative to a stationary frame, and \(u\) is the velocity of the same object relative to the stationary frame.
02

Applying the Condition \(v < c\)

Since \(v\) is given to be less than \(c\), we will apply this to the equation to see its effect. Replace \(v\) by some value \(k < c\), yielding: \(u' = \frac{u + k}{1 + \frac{uk}{c^2}}\).
03

Analyzing the Equation After Substitution

It's clear that the denominator of the equation, \(1 + \frac{uk}{c^2}\), will always be greater than 1 as long as \(u\) and \(k\) are real, positive, and less than \(c\). Therefore, \(u'\) must be less than both \(u\) and \(k\). Thus, \(u'\) will also be less than \(c\).

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