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University Physics Volume 3

5.6 Relativistic Velocity Transformation

University Physics Volume 35.6 Relativistic Velocity Transformation
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  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Derive the equations consistent with special relativity for transforming velocities in one inertial frame of reference into another.
  • Apply the velocity transformation equations to objects moving at relativistic speeds.
  • Examine how the combined velocities predicted by the relativistic transformation equations compare with those expected classically.

Remaining in place in a kayak in a fast-moving river takes effort. The river current pulls the kayak along. Trying to paddle against the flow can move the kayak upstream relative to the water, but that only accounts for part of its velocity relative to the shore. The kayak’s motion is an example of how velocities in Newtonian mechanics combine by vector addition. The kayak’s velocity is the vector sum of its velocity relative to the water and the water’s velocity relative to the riverbank. However, the relativistic addition of velocities is quite different.

Velocity Transformations

Imagine a car traveling at night along a straight road, as in Figure 5.19. The driver sees the light leaving the headlights at speed c within the car’s frame of reference. If the Galilean transformation applied to light, then the light from the car’s headlights would approach the pedestrian at a speed u=v+c,u=v+c, contrary to Einstein’s postulates.

An illustration of a car moving with velocity v, with light coming from the headlights at a greater velocity c.
Figure 5.19 According to experimental results and the second postulate of relativity, light from the car’s headlights moves away from the car at speed c and toward the observer on the sidewalk at speed c.

Both the distance traveled and the time of travel are different in the two frames of reference, and they must differ in a way that makes the speed of light the same in all inertial frames. The correct rules for transforming velocities from one frame to another can be obtained from the Lorentz transformation equations.

Relativistic Transformation of Velocity

Suppose an object P is moving at constant velocity u=(ux,uy,uz)u=(ux,uy,uz) as measured in the SS frame. The SS frame is moving along its x-axisx-axis at velocity v. In an increment of time dtdt, the particle is displaced by dxdx along the x-axis.x-axis. Applying the Lorentz transformation equations gives the corresponding increments of time and displacement in the unprimed axes:

dt=γ(dt+vdx/c2)dx=γ(dx+vdt)dy=dydz=dz.dt=γ(dt+vdx/c2)dx=γ(dx+vdt)dy=dydz=dz.

The velocity components of the particle seen in the unprimed coordinate system are then

dxdt=γ(dx+vdt)γ(dt+vdx/c2)=dxdt+v1+vc2dxdtdydt=dyγ(dt+vdx/c2)=dydtγ(1+vc2dxdt)dzdt=dzγ(dt+vdx/c2)=dzdtγ(1+vc2dxdt).dxdt=γ(dx+vdt)γ(dt+vdx/c2)=dxdt+v1+vc2dxdtdydt=dyγ(dt+vdx/c2)=dydtγ(1+vc2dxdt)dzdt=dzγ(dt+vdx/c2)=dzdtγ(1+vc2dxdt).

We thus obtain the equations for the velocity components of the object as seen in frame S:

ux=(ux+v1+vux/c2),uy=(uy/γ1+vux/c2),uz=(uz/γ1+vux/c2).ux=(ux+v1+vux/c2),uy=(uy/γ1+vux/c2),uz=(uz/γ1+vux/c2).

Compare this with how the Galilean transformation of classical mechanics says the velocities transform, by adding simply as vectors:

ux=ux+u,uy=uy,uz=uz.ux=ux+u,uy=uy,uz=uz.

When the relative velocity of the frames is much smaller than the speed of light, that is, when vc,vc, the special relativity velocity addition law reduces to the Galilean velocity law. When the speed v of SS relative to S is comparable to the speed of light, the relativistic velocity addition law gives a much smaller result than the classical (Galilean) velocity addition does.

Example 5.9

Velocity Transformation Equations for Light Suppose a spaceship heading directly toward Earth at half the speed of light sends a signal to us on a laser-produced beam of light (Figure 5.20). Given that the light leaves the ship at speed c as observed from the ship, calculate the speed at which it approaches Earth.

An illustration of a spaceship moving to the right with velocity v=0.500c and emitting a horizontal laser beam, which propagates to the right with velocity c.
Figure 5.20 How fast does a light signal approach Earth if sent from a spaceship traveling at 0.500c?

Strategy Because the light and the spaceship are moving at relativistic speeds, we cannot use simple velocity addition. Instead, we determine the speed at which the light approaches Earth using relativistic velocity addition.

Solution

  1. Identify the knowns: v=0.500c;u=c.v=0.500c;u=c.
  2. Identify the unknown: u.
  3. Express the answer as an equation: u=v+u1+vuc2.u=v+u1+vuc2.
  4. Do the calculation:
    u=v+u1+vuc2=0.500c+c1+(0.500c)(c)c2=(0.500+1)c(c2+0.500c2c2)=c.u=v+u1+vuc2=0.500c+c1+(0.500c)(c)c2=(0.500+1)c(c2+0.500c2c2)=c.

SignificanceRelativistic velocity addition gives the correct result. Light leaves the ship at speed c and approaches Earth at speed c. The speed of light is independent of the relative motion of source and observer, whether the observer is on the ship or earthbound.

Velocities cannot add to greater than the speed of light, provided that v is less than c and uu does not exceed c. The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.

Example 5.10

Relativistic Package Delivery Suppose the spaceship in the previous example approaches Earth at half the speed of light and shoots a canister at a speed of 0.750c (Figure 5.21). (a) At what velocity does an earthbound observer see the canister if it is shot directly toward Earth? (b) If it is shot directly away from Earth?

The first illustration shows the spaceship moving to the right, toward the earth, with velocity v=0.500c, and a canister moving to the right with a velocity u prime = 0.750c. The second illustration shows the spaceship moving to the right, toward the earth, with velocity v=0.500c, and a canister moving to the left with a velocity u prime = -0.750c.
Figure 5.21 A canister is fired at 0.7500c toward Earth or away from Earth.

Strategy Because the canister and the spaceship are moving at relativistic speeds, we must determine the speed of the canister by an earthbound observer using relativistic velocity addition instead of simple velocity addition.

Solution for (a)

  1. Identify the knowns: v=0.500c;u=0.750c.v=0.500c;u=0.750c.
  2. Identify the unknown: u.
  3. Express the answer as an equation: u=v+u1+vuc2.u=v+u1+vuc2.
  4. Do the calculation:
    u=v+u1+vuc2=0.500c+0.750c1+(0.500c)(0.750c)c2=0.909c.u=v+u1+vuc2=0.500c+0.750c1+(0.500c)(0.750c)c2=0.909c.

Solution for (b)

  1. Identify the knowns: v=0.500c;u=−0.750c.v=0.500c;u=−0.750c.
  2. Identify the unknown: u.
  3. Express the answer as an equation: u=v+u1+vuc2.u=v+u1+vuc2.
  4. Do the calculation:
    u=v+u1+vuc2=0.500c+(−0.750c)1+(0.500c)(−0.750c)c2=−0.400c.u=v+u1+vuc2=0.500c+(−0.750c)1+(0.500c)(−0.750c)c2=−0.400c.

Significance The minus sign indicates a velocity away from Earth (in the opposite direction from v), which means the canister is heading toward Earth in part (a) and away in part (b), as expected. But relativistic velocities do not add as simply as they do classically. In part (a), the canister does approach Earth faster, but at less than the vector sum of the velocities, which would give 1.250c. In part (b), the canister moves away from Earth at a velocity of −0.400c,−0.400c, which is faster than the −0.250c expected classically. The differences in velocities are not even symmetric: In part (a), an observer on Earth sees the canister and the ship moving apart at a speed of 0.409c, and at a speed of 0.900c in part (b).

Check Your Understanding 5.6

Distances along a direction perpendicular to the relative motion of the two frames are the same in both frames. Why then are velocities perpendicular to the x-direction different in the two frames?

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