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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
By applying the Einstein Velocity Addition formula, it is shown that if an object moves at a speed \(v < c\) relative to one inertial reference frame, then its speed with respect to any other inertial frame will also be less than \(c\). This validates the universal speed limit, \(c\), in nature.

Step by step solution

01

Understanding Einstein velocity addition formula

The Einstein Velocity Addition formula states that if one inertial frame (S) moves with velocity \(v\) related to another frame (S') and an object moves with velocity \(u\'\) in S', then the velocity \(u\) of that object in S is given by: \(u = (v+u') / (1+ vu'/c^2)\). Here \(c\) represents the speed of light in a vacuum, \(u\) and \(u'\) are the speeds of the object in their respective frames, and \(v\) is the relative velocity between the frames.
02

Proving the premise

By substituting \(u' = c\) into the formula, the velocity of the object in the frame S is computed to be: \(u = (v+c) / (1+ vc/c^2) = c\), which shows that the speed of the object in another inertial frame is also equal to \(c\). Therefore, if \(v<c\), the speed of an object relative to any other inertial frame must also be less than \(c\).
03

Concluding the result

By applying the Einstein Velocity Addition principle, it is demonstrated that if an object moves at a speed \(v<c\) with respect to one inertial reference frame, then its speed must be less than \(c\) with respect to any other inertial frame. This conclusion is based on fundamental principles of relativity, which imply a universal speed limit for all physical entities.

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Most popular questions from this chapter

Time dilation is sometimes described by saying that "moving clocks run slow." In what sense is this true? In what sense does the statement violate the spirit of relativity?

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