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Chapter 38: Problem 42

Galaxy A is measured to be receding from us on Earth with a speed of \(0.3 c .\) Galaxy \(\mathrm{B}\), located in precisely the opposite direction, is also receding from us at the same speed. What recessional velocity will an observer on galaxy A measure (a) for our galaxy, and (b) for galaxy B?

Short Answer

Expert verified
From galaxy A: (a) Earth's velocity = 0.55c, (b) galaxy B's velocity = 0c.

Step by step solution

01

Understand the Problem

Two galaxies A and B are receding from Earth with a speed of 0.3c each, in exactly opposite directions. We need to find the recessional velocity for an observer on galaxy A for both our galaxy (Earth) and galaxy B.
02

Define the Relevant Equation

Use the relativistic velocity addition formula to find the apparent velocity from the perspective of galaxy A. The formula is: \[ v' = \frac{v + u}{1 + \frac{vu}{c^2}} \] where v is the velocity of galaxy B from Earth's perspective and u is the velocity of Earth from galaxy A's perspective.
03

Calculate Earth's Recessional Velocity from Galaxy A

Here, the velocity of Earth from galaxy A's perspective is the same as the velocity of galaxy A from Earth's perspective, which is 0.3c. Thus, \[ v_A = 0.3c \] Since Earth is measured from galaxy A, we use u=0.3c for Earth. Applying the relativistic velocity formula: \[ v' = \frac{0.3c + 0.3c}{1 + \frac{0.3c \cdot 0.3c}{c^2}} = \frac{0.6c}{1 + 0.09} = \frac{0.6c}{1.09} \approx 0.55c \] So, the recessional velocity of Earth as seen from galaxy A is approximately 0.55c.
04

Calculate Galaxy B's Recessional Velocity from Galaxy A

Galaxy B is observed from Earth as moving away at a velocity of -0.3c. Because galaxy A itself is moving at 0.3c relative to Earth: \[ v_B = -0.3c \] Applying the same relativistic velocity formula: \[ v' = \frac{0.3c + (-0.3c)}{1 + \frac{0.3c \cdot (-0.3c)}{c^2}} = \frac{0}{1 - 0.09} = 0 \] So, the recessional velocity of galaxy B as seen from galaxy A is 0c.

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Key Concepts

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

relativity
Relativity is the framework that describes how motion and speed are perceived differently depending on the observer鈥檚 frame of reference. In special relativity, theorized by Einstein, it is important to account for the constancy of the speed of light and the fact that all observers, regardless of their relative motion, will measure the speed of light as the same, approximately 300,000 km/s or 1c. This has profound implications, leading to phenomena such as time dilation and length contraction. These effects become very noticeable only at velocities close to the speed of light.
velocity addition
Classically, if two objects move towards or away from each other, their velocities simply add. However, in relativity, we must consider the effects of traveling at significant fractions of the speed of light. The relativistic velocity addition formula is used to calculate the observed velocity when two objects in motion look at each other. The formula is:

where \( v' \) is the velocity as seen by one observer, \( v \) and \( u \) are the velocities of the objects, and \( c \) is the speed of light. This formula ensures no object exceeds the speed of light when combining velocities. Understanding this equation is crucial for computations involving high speeds, such as the movement of galaxies or particles in accelerators.
recession velocity
Recession velocity refers to the speed at which an object moves away from another. In the context of galaxies, it is the rate at which they recede due to the expanding universe. When dealing with objects moving in opposite directions at high velocities, it is essential to use the relativistic velocity addition formula instead of simple arithmetic addition to determine the recessional speed. For example, if two galaxies A and B move away from the Earth at 0.3c each in opposite directions, an observer on galaxy A sees our galaxy receding at approximately 0.55c and galaxy B at 0c.
special relativity
Special relativity, proposed by Albert Einstein, revolutionized the understanding of motion and space-time. It dictates that the laws of physics are the same for all non-accelerating observers and introduces the concept that the speed of light is the same in all frames of reference. Key predictions include:
  • Time dilation: Moving clocks run slower.
  • Length contraction: Moving objects shorten along the direction of motion.
  • Relativity of simultaneity: Events that are simultaneous in one frame may not be in another.

Special relativity forms the basis for the relativistic velocity addition formula and is essential when objects move at considerable fractions of the speed of light, as seen in the movement of distant galaxies or subatomic particles.

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Most popular questions from this chapter

Quite apart from effects due to the Earth's rotational and orbital motion, a laboratory reference frame on the Earth is not an inertial frame, as required by a strict interpretation of special relativity. It is not inertial because a particle released from rest at the Earth's surface does not remain at rest; it falls! Often, however, the events in an experiment for which one needs special relativity happen so quickly that we can ignore effects duc to gravitational accclcration. Considcr, for cxamplc, a proton moving horizontally at speed \(v=0.992 c\) through a 10 -m-wide detector in a laboratory test chamber. (a) How long will the transit through that detector take? (b) How far does the proton fall vertically during this time lapse? (c) What do you conclude about the suitability of the laboratory as an inertial frame in this case?

An unstable high-energy particle is created in a collision inside a detector and leaves a track \(1.05 \mathrm{~mm}\) long before it decays while still in the detector. Its speed relative to the detector was \(0.992 c\). How long did the particle live as recorded in its rest frame?

(a) Find an equation for the unknown mass \(m\) of a particle if you know its momentum \(p\) and its kinetic energy \(K\). Show that this expression reduces to an expected result for nonrelativistic particle speeds. (b) Find the mass of a particle whose kinetic energy is \(K=55.0 \mathrm{MeV}\) and whose momentum is \(p=\) \(121 \mathrm{MeV} / \mathrm{c}\). Express your answer as a decimal fraction or multiple of the mass \(m_{\mathrm{e}}\) of the electron.

A particle moves with uniform speed \(v_{y}^{\prime}=\Delta y^{\prime} / \Delta t^{\prime}\) in the \(y^{\prime}\) direction with respect to a rocket frame that moves along the \(x\) axis of a laboratory frame. Find exprcssions for the \(x\) -componcnt and for the \(y\) -componcnt of the particle's velocity in the laboratory frame.

A meter stick lies at rest in the rocket frame and makes an angle \(\phi^{\prime}\) with the \(x^{\prime}\) axis as measured by the rocket observer. The laboratory observer measures the \(x\) - and \(y\) -components of the meter stick as it streaks past. From these components the laboratory observer computes the angle \(\phi\) that the stick makes with his \(x\) axis. (a) Find an expression for the angle \(\phi\) in terms of the angle \(\phi^{\prime}\) and the relative speed \(v^{\text {rel }}\) between rocket and laboratory frames. (b) What is the length of the "meter" stick measured by the laboratory observer? (c) Optional: Why is your expression in part (a) different from equations derived in Problems 34 and \(35 ?\)
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