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Chapter 35: Problem 40

Use the relativistic velocity addition to reconfirm that the speed of light with respect to any inertial reference frame is \(c\). Assume one-dimensional motion along a common \(x\) -axis.

Short Answer

Expert verified
Question: Use the relativistic velocity addition formula to confirm that the speed of light is the same in any inertial reference frame in a one-dimensional motion along the common x-axis. Answer: By applying the relativistic velocity addition formula and setting the velocity of the object relative to both observers as equal to the speed of light (c), we derived the equation \(c + v = v + c\). This result confirms that the speed of light remains constant in any inertial reference frame, regardless of the relative velocity (v).

Step by step solution

01

Write down the relativistic velocity addition formula

The relativistic velocity addition formula is given by: \(u = \frac{v +u'}{1 + \frac{vu'}{c^2}}\) where \(u\) is the velocity of an object relative to an observer, \(v\) is the relative velocity of two observers, and \(u'\) is the velocity of the object relative to the second observer. The speed of light is denoted by \(c\).
02

Set the speed of light as the velocity of the object relative to both observers

Since we want to show that the speed of light remains constant in any inertial reference frame, we will set the velocity of the object relative to both observers as equal to the speed of light, \(c\). Thus, \(u = c\) and \(u' = c\).
03

Substitute the values of \(u\), \(u'\) and \(c\) into the relativistic velocity addition formula

Substitute the values of \(u = c\) and \(u' = c\) into the relativistic velocity addition formula: \(c = \frac{v+c}{1 + \frac{vc}{c^2}}\) Now, we will simplify the equation to show that the speed of light remains constant.
04

Simplify the equation

First, cancel the \(c^2\) term in the denominator: \(c = \frac{v + c}{1 + \frac{v}{c}}\) Next, multiply both sides of the equation by the denominator to get rid of the fraction: \(c(1 + \frac{v}{c}) = v + c\) Distribute \(c\) through the parentheses: \(c + v = v + c\)
05

Verify the speed of light remains constant

As per the result, the equation simplifies to \(c + v = v + c\). This result implies that the speed of light remains constant in any inertial reference frame, regardless of the relative velocity \(v\). Since the speed of light is \(c\) with respect to both observers, the problem is solved and the exercise is complete.

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Most popular questions from this chapter

A proton with a momentum of \(3.0 \mathrm{GeV} / \mathrm{c}\) is moving with what velocity relative to the observer? a) \(0.31 c\) c) \(0.91 c\) e) \(3.2 c\) b) \(0.33 c\) d) \(0.95 c\)

A spaceship is traveling at two-thirds of the speed of light directly toward a stationary asteroid. If the spaceship turns on it headlights, what will be the speed of the light traveling from the spaceship to the asteroid as observed by a) someone on the spaceship? b) someone on the asteroid?

What is the relative speed between two objects approaching each other head on, if each is traveling at speed of \(0.600 c\) as measured by an observer on Earth?

Find the speed of light in feet per nanosecond, to three significant figures.

A particle of rest mass \(m_{0}\) travels at a speed \(v=0.20 c\) How fast must the particle travel in order for its momentum to increase to twice its original momentum? a) \(0.40 c\) c) \(0.38 c\) e) \(0.99 c\) b) \(0.10 c\) d) \(0.42 c\)
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