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Chapter 2: Problem 17

Observer \(O\) fires a light beam in the \(y\) direction \(\left(v_{y}=c\right)\). Use the Lorentz velocity transformation to find \(v_{x}^{\prime}\) and \(v_{1}^{\prime}\) and show that \(O^{\prime}\) also measures the value \(c\) for the speed of light. Assume \(O^{\prime}\) moves relative to \(O\) with velocity \(u\) in the \(x\) direction.

Short Answer

Expert verified
Observer O' measures the light beam鈥檚 speed as c.

Step by step solution

01

Understand the Transformation Context

Given the observer O fires a light beam in the y direction with the velocity of light, and observer O' moves relative to O with velocity u in the x direction. The Lorentz velocity transformations are used to transform the velocities from one frame to another.
02

Identify the Given Variables

The light beam has velocities \( v_{y} = c \) and \( v_{x} = 0 \) in frame O. The relative velocity between the frames is u in the x direction.
03

Lorentz Velocity Transformation for v_x'

The transformation for the x-component of velocity is: \[ v_{x}^{\text{'} } = \frac{ v_{x} - u }{ 1 - \frac{ u \cdot v_{x} }{ c^{2} } } \] Substituting \( v_{x} = 0 \): \[ v_{x}^{\text{'} } = \frac{ 0 - u }{ 1 - \frac{ u \cdot 0 }{ c^{2} } } = -u \]
04

Lorentz Velocity Transformation for v_y'

The transformation for the y-component of velocity is: \[ v_{y}^{\text{'} } = \frac{ v_{y} \sqrt{ 1 - \frac{ u^{2} }{ c^{2} } } }{ 1 - \frac{ u \cdot v_{x} }{ c^{2} } } \] Substituting \( v_{y} = c \) and \( v_{x} = 0 \): \[ v_{ y }^{ \text{'} } = \frac{ c \sqrt{ 1 - \frac{ u^{ 2 } }{ c^{ 2 } } } }{ 1 - 0 } = c \sqrt{ 1 - \frac{ u^{ 2 } }{ c^{ 2 } } } \]
05

Calculate the Total Speed in O'

Using the relationship for speed in frame O': \[ v^{ \text{'} } = \sqrt{ ( v_{ x }^{ \text{'} } )^{2} + ( v_{ y }^{ \text{'} } )^{2} } \] Substitute the results: \[ v^{ \text{'} } = \sqrt{ ( -u )^{2} + ( c \sqrt{ 1 - \frac{ u^{ 2} }{ c^{ 2} } } )^{ 2 } } \] Where simplifying it becomes: \[ v^{ \text{'} } = \sqrt{ u^{ 2 } + c^{ 2 } ( 1 - \frac{ u^{ 2} }{ c^{ 2 } } ) } = \sqrt{ u^{ 2 } + c^{ 2 } - u^{ 2 } } = \sqrt{ c^{ 2 } } = c \] Observer O' measures the light beam鈥檚 speed as c.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lorentz transformation
The Lorentz transformation is crucial in understanding how measurements of space and time differ for observers in different inertial frames of motion. These transformations, formulated by Hendrik Lorentz, are a key component of Einstein's theory of Special Relativity.
They help us convert the coordinates of one observer moving at a constant speed relative to another observer. When applying the Lorentz transformation to velocities, we can determine how the velocity of an object changes from one frame to another.
The formulas focus on the x and y components of velocity, taking into account the relative motion between observers and ensuring the speed of light remains constant regardless of the frame of reference.
Relative velocity
Relative velocity describes how the velocity of an object changes based on the observer's frame of reference. When two observers are in different frames of motion, they may measure different velocities for the same object.
In the context of the Lorentz velocity transformation, we look at how velocity components change. An observer moving at a constant velocity u in the x direction will see different velocity components than a stationary observer. For example, if Observer O sees a light beam moving vertically, Observer O' who is moving in the x direction will measure different horizontal and vertical velocity components for the same beam.
This is because the Lorentz transformations adjust velocities such that the speed of light remains constant for all observers.
Speed of light
The speed of light, denoted by c, is a fundamental constant in physics, measuring approximately 299,792,458 meters per second. This constancy is a cornerstone of the theory of Special Relativity.
No matter the relative motion of the observers, the speed of light remains unchanged at c. This principle leads to several counterintuitive results, but it has been repeatedly confirmed by experimental evidence.
In our example, even if Observer O' is moving relative to Observer O, they will both measure the speed of a light beam as c. The Lorentz velocity transformation ensures that the calculations adjust the observed velocities so that this constancy holds true.
Special relativity
Special Relativity, introduced by Albert Einstein in 1905, revolutionized our understanding of space, time and motion. It rests on two foundational principles: the constancy of the speed of light and the laws of physics being the same in all inertial frames.
This theory challenges concepts from classical mechanics, particularly when dealing with objects moving at speeds close to the speed of light. In our exercise scenario, even though Observer O' moves relative to Observer O, the speed of light that both measure remains consistent at c.
This is a direct consequence of special relativity, emphasizing that observations of spatial and temporal intervals differ between observers in relative motion, but the speed of light remains a universal invariant.

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Most popular questions from this chapter

Suppose the speed of light were \(1000 \mathrm{mi} / \mathrm{h} .\) You are traveling on a flight from Los Angeles to Boston, a distance of 3000 mi. The plane's speed is a constant \(600 \mathrm{mi} / \mathrm{h}\). You leave Los Angeles at 10: 00 am, as indicated by your wristwatch and by a clock in the airport. \((a)\) According to your watch, what time is it when you land in Boston? \((b)\) In the Boston airport is a clock that is synchronized to read exactly the same time as the clock in the Los Angeles airport. What time does that clock read when you land in Boston? \((c)\) The following day when the Boston clock that records Los Angeles time reads 10: 00 am, you leave Boston to return to Los Angeles on the same airplane. When you land in Los Angeles, what are the times read on your watch and on the airport clock?

An electron is moving at a speed of \(0.85 c .\) By how much must its kinetic energy increase to raise its speed to \(0.91 \mathrm{c} ?\)

Agnes makes a round trip at a constant speed to a star that is 16 light-years distant from Earth, while twin brother Bert remains on Earth. When Agnes returns to Earth, she reports that she has celebrated 20 birthdays during her journey. (a) What was her speed during her journey? (b) How old is Bert when she returns?

Suppose Amelia traveled at a speed of \(0.80 c\) to a star that (according to Casper on Earth) is 8.0 light-years away. Casper ages 20 years during Amelia's round trip. How much younger than Casper is Amelia when she returns to Earth?

Two spaceships approach the Earth from opposite directions. According to an observer on the Earth, ship \(A\) is moving at a speed of \(0.743 c\) and ship \(B\) at a speed of \(0.831 c .\) What is the velocity of ship \(A\) as observed from ship \(B ?\) Of ship \(B\) as observed from ship \(A\) ?
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