If you’ve ever seen a kayak move down a fast-moving river, you know that remaining in the same place would be hard. The river current pulls the kayak along. Pushing the oars back against the water can move the kayak forward in the water, but that only accounts for part of the velocity. The kayak’s motion is an example of classical addition of velocities. In classical physics, velocities add as vectors. The kayak’s velocity is the vector sum of its velocity relative to the water and the water’s velocity relative to the riverbank.

# Classical Velocity Addition

For simplicity, we restrict our consideration of velocity addition to one-dimensional motion. Classically, velocities add like regular numbers in one-dimensional motion. (See [link].) Suppose, for example, a girl is riding in a sled at a speed 1.0 m/s relative to an observer. She throws a snowball first forward, then backward at a speed of 1.5 m/s relative to the sled. We denote direction with plus and minus signs in one dimension; in this example, forward is positive. Let $v$ be the velocity of the sled relative to the Earth, $u$ the velocity of the snowball relative to the Earth-bound observer, and $u\prime $ the velocity of the snowball relative to the sled.

Thus, when the girl throws the snowball forward, $u=\mathrm{1.0\; m/s}+\mathrm{1.5\; m/s}=\mathrm{2.5\; m/s}$. It makes good intuitive sense that the snowball will head towards the Earth-bound observer faster, because it is thrown forward from a moving vehicle. When the girl throws the snowball backward, $u=\mathrm{1.0\; m/s}+(-\mathrm{1.5\; m/s})=-\mathrm{0.5\; m/s}$. The minus sign means the snowball moves away from the Earth-bound observer.

# Relativistic Velocity Addition

The second postulate of relativity (verified by extensive experimental observation) says that classical velocity addition does not apply to light. Imagine a car traveling at night along a straight road, as in [link]. If classical velocity addition applied to light, then the light from the car’s headlights would approach the observer on the sidewalk at a speed $\text{u=v+c}$. But we know that light will move away from the car at speed $c$ relative to the driver of the car, and light will move towards the observer on the sidewalk at speed $c$, too.

Suppose a spaceship heading directly towards the Earth at half the speed of light sends a signal to us on a laser-produced beam of light. Given that the light leaves the ship at speed $c$ as observed from the ship, calculate the speed at which it approaches the Earth.

**Strategy**

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

**Solution**

- Identify the knowns. $\text{v=}0\text{.}\text{500}c$; $u\prime =c$
- Identify the unknown. $u$
- Choose the appropriate equation. $u=\frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}$
- Plug the knowns into the equation.
$\begin{array}{lll}u& =& \frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}\\ & =& \frac{\text{0.500}c+c}{1+\frac{(\text{0.500}c)(c)}{{c}^{2}}}\\ & =& \frac{(\text{0.500}+1)c}{1+\frac{\text{0.500}{c}^{2}}{{c}^{2}}}\\ & =& \frac{\text{1.500}c}{1+\text{0.500}}\\ & =& \frac{\text{1.500}c}{\text{1.500}}\\ & =& c\end{array}$

**Discussion**

Relativistic velocity addition gives the correct result. Light leaves the ship at speed $c$ and approaches the 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 Earth-bound.

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

Suppose the spaceship in the previous example is approaching the Earth at half the speed of light and shoots a canister at a speed of $0.750c$. (a) At what velocity will an Earth-bound observer see the canister if it is shot directly towards the Earth? (b) If it is shot directly away from the Earth? (See [link].)

**Strategy**

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

**Solution for (a)**

- Identify the knowns. $\text{v=}0.500c$; $u\prime =0\text{.}\text{750}c$
- Identify the unknown. $u$
- Choose the appropriate equation. $\text{u=}\frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}$
- Plug the knowns into the equation.
$\begin{array}{ll}u& =& \frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}\\ & =& \frac{0.500\text{c +}0.750c}{1+\frac{(0.500c)(0.750c)}{{c}^{2}}}\\ & =& \frac{1.250c}{1+0.375}\\ & =& 0.909c\end{array}$

**Solution for (b)**

- Identify the knowns. $\text{v}=0.500c$; $u\prime =-0.750c$
- Identify the unknown. $u$
- Choose the appropriate equation. $\text{u}=\frac{\mathrm{v+u}\prime}{1+\frac{v\text{u}\prime}{{c}^{2}}}$
- Plug the knowns into the equation.
$\begin{array}{ll}u& =& \frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}\\ & =& \frac{0.500\mathrm{c\; +}(-0.750c)}{1+\frac{(0.500c)(-0.750c)}{{c}^{2}}}\\ & =& \frac{-0.250c}{1-0.375}\\ & =& -0.400c\end{array}$

**Discussion**

The minus sign indicates velocity away from the Earth (in the opposite direction from $v$), which means the canister is heading towards the 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 the Earth faster, but not at the simple sum of $1.250c$. The total velocity is less than you would get classically. And in part (b), the canister moves away from the Earth at a velocity of $-0.400c$, which is * faster* than the $\mathrm{-0.250}c$ you would expect classically. The velocities are not even symmetric. In part (a) the canister moves $0.409c$ faster than the ship relative to the Earth, whereas in part (b) it moves $0.900c$ slower than the ship.

# Doppler Shift

Although the speed of light does not change with relative velocity, the frequencies and wavelengths of light do. First discussed for sound waves, a Doppler shift occurs in any wave when there is relative motion between source and observer.

In the Doppler equation, ${\lambda}_{\text{obs}}$ is the observed wavelength, ${\lambda}_{s}$ is the source wavelength, and $u$ is the relative velocity of the source to the observer. The velocity $u$ is positive for motion away from an observer and negative for motion toward an observer. In terms of source frequency and observed frequency, this equation can be written

Notice that the – and + signs are different than in the wavelength equation.

Suppose a galaxy is moving away from the Earth at a speed $\text{0.825}c$ . It emits radio waves with a wavelength of $0\text{.}\text{525}\phantom{\rule{0.25em}{0ex}}\text{m}$. What wavelength would we detect on the Earth?

**Strategy**

Because the galaxy is moving at a relativistic speed, we must determine the Doppler shift of the radio waves using the relativistic Doppler shift instead of the classical Doppler shift.

**Solution**

- Identify the knowns. $\text{u=}0\text{.}\text{825}c$ ; ${\lambda}_{s}=0\text{.}\text{525}\phantom{\rule{0.25em}{0ex}}m$
- Identify the unknown. ${\lambda}_{\text{obs}}$
- Choose the appropriate equation. ${\lambda}_{\text{obs}}{\text{=\lambda}}_{s}\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}$
- Plug the knowns into the equation.
$\begin{array}{lll}{\lambda}_{\text{obs}}& =& {\text{\lambda}}_{s}\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}\\ & =& (\mathrm{0.525\; m})\sqrt{\frac{1+\frac{0\text{.}\text{825}\text{c}}{c}}{1-\frac{0\text{.}\text{825}\text{c}}{c}}}\\ & =& \text{1.70 m.}\end{array}$

**Discussion**

Because the galaxy is moving away from the Earth, we expect the wavelengths of radiation it emits to be redshifted. The wavelength we calculated is 1.70 m, which is redshifted from the original wavelength of 0.525 m.

The relativistic Doppler shift is easy to observe. This equation has everyday applications ranging from Doppler-shifted radar velocity measurements of transportation to Doppler-radar storm monitoring. In astronomical observations, the relativistic Doppler shift provides velocity information such as the motion and distance of stars.

# Section Summary

- With classical velocity addition, velocities add like regular numbers in one-dimensional motion: $\text{u=v+u}\prime $, where $v$ is the velocity between two observers, $u$ is the velocity of an object relative to one observer, and $u\prime $ is the velocity relative to the other observer.
- Velocities cannot add to be greater than the speed of light. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed:
$\text{u=}\frac{\text{v+u}\prime}{1+\frac{v\text{u}\prime}{{c}^{2}}}$
- An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation
${\lambda}_{\text{obs}}{\text{=\lambda}}_{s}\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}$${\lambda}_{\text{obs}}$ is the observed wavelength, ${\lambda}_{s}$ is the source wavelength, and $u$ is the relative velocity of the source to the observer.

# Conceptual Questions

Explain the meaning of the terms “red shift” and “blue shift” as they relate to the relativistic Doppler effect.

What happens to the relativistic Doppler effect when relative velocity is zero? Is this the expected result?

Is the relativistic Doppler effect consistent with the classical Doppler effect in the respect that ${\lambda}_{\text{obs}}$ is larger for motion away?

All galaxies farther away than about $\text{50}\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{ly}$ exhibit a red shift in their emitted light that is proportional to distance, with those farther and farther away having progressively greater red shifts. What does this imply, assuming that the only source of red shift is relative motion? (Hint: At these large distances, it is space itself that is expanding, but the effect on light is the same.)

# Problems & Exercises

Suppose a spaceship heading straight towards the Earth at $0\text{.}\text{750}c$ can shoot a canister at $0\text{.}\text{500}c$ relative to the ship. (a) What is the velocity of the canister relative to the Earth, if it is shot directly at the Earth? (b) If it is shot directly away from the Earth?

(a) $0\text{.}\text{909}c$

(b) $0\text{.}\text{400}c$

Repeat the previous problem with the ship heading directly away from the Earth.

If a spaceship is approaching the Earth at $0.100c$ and a message capsule is sent toward it at $0.100c$ relative to the Earth, what is the speed of the capsule relative to the ship?

$0\text{.}\text{198}c$

(a) Suppose the speed of light were only $\text{3000 m/s}$. A jet fighter moving toward a target on the ground at $\text{800 m/s}$ shoots bullets, each having a muzzle velocity of $\text{1000 m/s}$. What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.

If a galaxy moving away from the Earth has a speed of $\mathrm{1000\; km/s}$ and emits $\text{656 nm}$ light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on the Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of the Earth in its orbit negligible here?

a) $\text{658 nm}$

b) red

c) $v/\text{c}=9\text{.}\text{92}\times {\text{10}}^{-5}$ (negligible)

A space probe speeding towards the nearest star moves at $0\text{.}\text{250}c$ and sends radio information at a broadcast frequency of 1.00 GHz. What frequency is received on the Earth?

If two spaceships are heading directly towards each other at $0\text{.}\text{800}c$, at what speed must a canister be shot from the first ship to approach the other at $0\text{.}\text{999}c$ as seen by the second ship?

$0\text{.}\text{991}c$

Two planets are on a collision course, heading directly towards each other at $0\text{.}\text{250}c$. A spaceship sent from one planet approaches the second at $0\text{.}\text{750}c$ as seen by the second planet. What is the velocity of the ship relative to the first planet?

When a missile is shot from one spaceship towards another, it leaves the first at $0\text{.}\text{950}c$ and approaches the other at $0\text{.}\text{750}c$. What is the relative velocity of the two ships?

$-0\text{.}\text{696}c$

What is the relative velocity of two spaceships if one fires a missile at the other at $0.750c$ and the other observes it to approach at $0.950c$?

Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?

$0\text{.}\text{01324}c$

A highway patrol officer uses a device that measures the speed of vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar has a frequency of 100 GHz and the returning echo has a frequency 15.0 kHz higher. What is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain not to round off until the end of the problem, because the effect is small.

Prove that for any relative velocity $v$ between two observers, a beam of light sent from one to the other will approach at speed $c$ (provided that $v$ is less than $c$, of course).

$u\prime \phantom{\rule{0.25em}{0ex}}=c$, so

$\begin{array}{ll}u& =& \frac{\text{v+u}\prime}{1+(\text{vu}\prime /{c}^{2})}=\frac{\text{v+c}}{1+(\text{vc}/{c}^{2})}=\frac{\text{v+c}}{1+(v/c)}\\ & =& \frac{c(\text{v+c})}{\text{c+v}}=c\end{array}$

Show that for any relative velocity $v$ between two observers, a beam of light projected by one directly away from the other will move away at the speed of light (provided that $v$ is less than $c$, of course).

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $\text{12}\text{.}0\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{ly}$ ly away is receding from us at 0.$0.900c$, at what velocity relative to us must we send an exploratory probe to approach the other galaxy at $0.990c$, as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)

a) $0\text{.}\text{99947}c$

b) $1\text{.}\text{2064}\times {\text{10}}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{y}$

c) $1\text{.}\text{2058}\times {\text{10}}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{y}$ (all to sufficient digits to show effects)

- College Physics
- Preface
- Introduction: The Nature of Science and Physics
- Kinematics
- Introduction to One-Dimensional Kinematics
- Displacement
- Vectors, Scalars, and Coordinate Systems
- Time, Velocity, and Speed
- Acceleration
- Motion Equations for Constant Acceleration in One Dimension
- Problem-Solving Basics for One-Dimensional Kinematics
- Falling Objects
- Graphical Analysis of One-Dimensional Motion

- Two-Dimensional Kinematics
- Dynamics: Force and Newton's Laws of Motion
- Introduction to Dynamics: Newton’s Laws of Motion
- Development of Force Concept
- Newton’s First Law of Motion: Inertia
- Newton’s Second Law of Motion: Concept of a System
- Newton’s Third Law of Motion: Symmetry in Forces
- Normal, Tension, and Other Examples of Forces
- Problem-Solving Strategies
- Further Applications of Newton’s Laws of Motion
- Extended Topic: The Four Basic Forces—An Introduction

- Further Applications of Newton's Laws: Friction, Drag, and Elasticity
- Uniform Circular Motion and Gravitation
- Work, Energy, and Energy Resources
- Linear Momentum and Collisions
- Statics and Torque
- Rotational Motion and Angular Momentum
- Introduction to Rotational Motion and Angular Momentum
- Angular Acceleration
- Kinematics of Rotational Motion
- Dynamics of Rotational Motion: Rotational Inertia
- Rotational Kinetic Energy: Work and Energy Revisited
- Angular Momentum and Its Conservation
- Collisions of Extended Bodies in Two Dimensions
- Gyroscopic Effects: Vector Aspects of Angular Momentum

- Fluid Statics
- Fluid Dynamics and Its Biological and Medical Applications
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- Flow Rate and Its Relation to Velocity
- Bernoulli’s Equation
- The Most General Applications of Bernoulli’s Equation
- Viscosity and Laminar Flow; Poiseuille’s Law
- The Onset of Turbulence
- Motion of an Object in a Viscous Fluid
- Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Temperature, Kinetic Theory, and the Gas Laws
- Heat and Heat Transfer Methods
- Thermodynamics
- Introduction to Thermodynamics
- The First Law of Thermodynamics
- The First Law of Thermodynamics and Some Simple Processes
- Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- Applications of Thermodynamics: Heat Pumps and Refrigerators
- Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

- Oscillatory Motion and Waves
- Introduction to Oscillatory Motion and Waves
- Hooke’s Law: Stress and Strain Revisited
- Period and Frequency in Oscillations
- Simple Harmonic Motion: A Special Periodic Motion
- The Simple Pendulum
- Energy and the Simple Harmonic Oscillator
- Uniform Circular Motion and Simple Harmonic Motion
- Damped Harmonic Motion
- Forced Oscillations and Resonance
- Waves
- Superposition and Interference
- Energy in Waves: Intensity

- Physics of Hearing
- Electric Charge and Electric Field
- Introduction to Electric Charge and Electric Field
- Static Electricity and Charge: Conservation of Charge
- Conductors and Insulators
- Coulomb’s Law
- Electric Field: Concept of a Field Revisited
- Electric Field Lines: Multiple Charges
- Electric Forces in Biology
- Conductors and Electric Fields in Static Equilibrium
- Applications of Electrostatics

- Electric Potential and Electric Field
- Electric Current, Resistance, and Ohm's Law
- Circuits, Bioelectricity, and DC Instruments
- Magnetism
- Introduction to Magnetism
- Magnets
- Ferromagnets and Electromagnets
- Magnetic Fields and Magnetic Field Lines
- Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- Force on a Moving Charge in a Magnetic Field: Examples and Applications
- The Hall Effect
- Magnetic Force on a Current-Carrying Conductor
- Torque on a Current Loop: Motors and Meters
- Magnetic Fields Produced by Currents: Ampere’s Law
- Magnetic Force between Two Parallel Conductors
- More Applications of Magnetism

- Electromagnetic Induction, AC Circuits, and Electrical Technologies
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Induced Emf and Magnetic Flux
- Faraday’s Law of Induction: Lenz’s Law
- Motional Emf
- Eddy Currents and Magnetic Damping
- Electric Generators
- Back Emf
- Transformers
- Electrical Safety: Systems and Devices
- Inductance
- RL Circuits
- Reactance, Inductive and Capacitive
- RLC Series AC Circuits

- Electromagnetic Waves
- Geometric Optics
- Vision and Optical Instruments
- Wave Optics
- Introduction to Wave Optics
- The Wave Aspect of Light: Interference
- Huygens's Principle: Diffraction
- Young’s Double Slit Experiment
- Multiple Slit Diffraction
- Single Slit Diffraction
- Limits of Resolution: The Rayleigh Criterion
- Thin Film Interference
- Polarization
- *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- Special Relativity
- Introduction to Quantum Physics
- Atomic Physics
- Introduction to Atomic Physics
- Discovery of the Atom
- Discovery of the Parts of the Atom: Electrons and Nuclei
- Bohr’s Theory of the Hydrogen Atom
- X Rays: Atomic Origins and Applications
- Applications of Atomic Excitations and De-Excitations
- The Wave Nature of Matter Causes Quantization
- Patterns in Spectra Reveal More Quantization
- Quantum Numbers and Rules
- The Pauli Exclusion Principle

- Radioactivity and Nuclear Physics
- Medical Applications of Nuclear Physics
- Particle Physics
- Frontiers of Physics
- Atomic Masses
- Selected Radioactive Isotopes
- Useful Information
- Glossary of Key Symbols and Notation