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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

An astronaut on a spaceship traveling at a speed of \(0.50 c\) is holding a meter stick parallel to the direction of motion. a) What is the length of the meter stick as measured by another astronaut on the spaceship? b) If an observer on Earth could observe the meter stick, what would be the length of the meter stick as measured by that observer?

Two stationary space stations are separated by a distance of \(100 .\) light- years, as measured by someone on one of the space stations. A spaceship traveling at \(0.950 c\) relative to the space stations passes by one of the space stations heading directly toward the other one. How long will it take to reach the other space station, as measured by someone on the spaceship? How much time will have passed for a traveler on the spaceship as it travels from one space station to the other, as measured by someone on one of the space stations? Round the answers to the nearest year.

In your physics class you have just learned about the relativistic frequency shift, and you decide to amaze your friends at a party. You tell them that once you drove through a stop light and that when you were pulled over you did not get ticketed because you explained to the police officer that the relativistic Doppler shift made the red light of wavelength \(650 \mathrm{nm}\) appear green to you, with a wavelength of \(520 \mathrm{nm}\). If your story had been true, how fast would you have been traveling?

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) \(0.95 c\)

The hot filament of the electron gun in a cathode ray tube releases electrons with nearly zero kinetic energy. The electrons are next accelerated under a potential difference of \(5.00 \mathrm{kV}\), before being steered toward the phosphor on the screen of the tube. a) Calculate the kinetic energy acquired by the electron under this accelerating potential difference. b) Is the electron moving at relativistic speed? c) What is the electron's total energy and momentum? (Give both values, relativistic and nonrelativistic, for both quantities.)
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