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Chapter 26: Problem 25

Kurt is measuring the speed of light in an evacuated chamber aboard a spaceship traveling with a constant velocity of \(0.60 c\) with respect to Earth. The light is moving in the direction of motion of the spaceship. Siu- Ling is on Earth watching the experiment. With what speed does the light in the vacuum chamber travel, according to Siu-Ling's observations?

Short Answer

Expert verified
Answer: The observed speed of light according to Siu-Ling's observations is c (the speed of light in a vacuum).

Step by step solution

01

Principle of relativistic velocity addition

The principle of relativistic addition of velocities states that if object A is moving at velocity \(v_A\) relative to object B, and object B is moving at velocity \(v_B\) relative to object C, then the velocity of object A relative to object C, \(v_{AC}\), can be calculated using the formula: \(v_{AC} = \frac{v_A + v_B}{1 + \frac{v_A v_B}{c^2}}\) In our case, the spaceship (object B) is moving with a velocity of \(0.60c\) with respect to Earth (object C), and the light (object A) is moving with a velocity \(c\) with respect to the spaceship. We want to find the velocity of the light (object A) with respect to Earth (object C), which is \(v_{AC}\).
02

Substitute the given values into the formula

Now, we will substitute the given values into the formula for relativistic addition of velocities: \(v_A = c\), \(v_B = 0.60c\), and \(c\) is the speed of light in a vacuum. \(v_{AC} = \frac{c + 0.60c}{1 + \frac{c(0.60c)}{c^2}}\)
03

Simplify the expression for \(v_{AC}\)

Let's simplify the expression: \(v_{AC} = \frac{1.60c}{1 + 0.60} = \frac{1.60c}{1.60}\)
04

Calculate the final result for the speed of light according to Siu-Ling's observations

Now we can find \(v_{AC}\): \(v_{AC} = \frac{1.60c}{1.60} = c\) The speed of light according to Siu-Ling's observations is \(c\), the same as the speed of light in a vacuum. This is expected because, according to the principles of special relativity, the speed of light is the same for all observers, regardless of their relative motion.

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Most popular questions from this chapter

An extremely relativistic particle is one whose kinetic energy is much larger than its rest energy. Show that for an extremely relativistic particle $E \approx p c$.
A futuristic train moving in a straight line with a uniform speed of \(0.80 c\) passes a series of communications towers. The spacing between the towers, according to an observer on the ground, is \(3.0 \mathrm{km} .\) A passenger on the train uses an accurate stopwatch to see how often a tower passes him. (a) What is the time interval the passenger measures between the passing of one tower and the next? (b) What is the time interval an observer on the ground measures for the train to pass from one tower to the next?
A spaceship travels toward Earth at a speed of \(0.97 c .\) The occupants of the ship are standing with their torsos parallel to the direction of travel. According to Earth observers, they are about \(0.50 \mathrm{m}\) tall and $0.50 \mathrm{m}$ wide. What are the occupants' (a) height and (b) width according to others on the spaceship?
Octavio, traveling at a speed of \(0.60 c,\) passes Tracy and her barn. Tracy, who is at rest with respect to her barn, says that the barn is 16 m long in the direction in which Octavio is traveling, \(4.5 \mathrm{m}\) high, and $12 \mathrm{m}$ deep. (a) What does Tracy say is the volume of her barn? (b) What volume does Octavio measure?
A plane trip lasts \(8.0 \mathrm{h} ;\) the average speed of the plane during the flight relative to Earth is \(220 \mathrm{m} / \mathrm{s}\). What is the time difference between an atomic clock on board the plane and one on the ground, assuming they were synchronized before the flight? (Ignore general relativistic complications due to gravity and the acceleration of the plane.)
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