Chapter 28 Special Relativity
28.4 Relativistic Addition of Velocities
Summary
- Calculate relativistic velocity addition.
- Explain when relativistic velocity addition should be used instead of classical addition of velocities.
- Calculate relativistic Doppler shift.

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 Figure 2.) 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

Classical Velocity Addition
Thus, when the girl throws the snowball forward,
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 Figure 3. 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

Relativistic Velocity Addition
Either light is an exception, or the classical velocity addition formula only works at low velocities. The latter is the case. The correct formula for one-dimensional relativistic velocity addition is
where
Example 1: Showing that the Speed of Light towards an Observer is Constant (in a Vacuum): The Speed of Light is the Speed of Light
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

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.
; - Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
Discussion
Relativistic velocity addition gives the correct result. Light leaves the ship at speed
Velocities cannot add to greater than the speed of light, provided that
Example 2: Comparing the Speed of Light towards and away from an Observer: Relativistic Package Delivery
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

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.
; - Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
Solution for (b)
- Identify the knowns.
; - Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
Discussion
The minus sign indicates velocity away from the Earth (in the opposite direction from
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.
Relativistic Doppler Effects
The observed wavelength of electromagnetic 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 towards the observer.
In the Doppler equation,
Notice that the – and + signs are different than in the wavelength equation.
Career Connection: Astronomer
If you are interested in a career that requires a knowledge of special relativity, there’s probably no better connection than astronomy. Astronomers must take into account relativistic effects when they calculate distances, times, and speeds of black holes, galaxies, quasars, and all other astronomical objects. To have a career in astronomy, you need at least an undergraduate degree in either physics or astronomy, but a Master’s or doctoral degree is often required. You also need a good background in high-level mathematics.
Example 3: Calculating a Doppler Shift: Radio Waves from a Receding Galaxy
Suppose a galaxy is moving away from the Earth at a speed
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.
; - Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
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.
Check Your Understanding
1: Suppose a space probe moves away from the Earth at a speed
Section Summary
- With classical velocity addition, velocities add like regular numbers in one-dimensional motion:
, where is the velocity between two observers, is the velocity of an object relative to one observer, and 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:
- 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
is the observed wavelength, is the source wavelength, and is the relative velocity of the source to the observer.
Conceptual Questions
1: Explain the meaning of the terms “red shift” and “blue shift” as they relate to the relativistic Doppler effect.
2: What happens to the relativistic Doppler effect when relative velocity is zero? Is this the expected result?
3: Is the relativistic Doppler effect consistent with the classical Doppler effect in the respect that
4: All galaxies farther away than about
Problems & Exercises
1: Suppose a spaceship heading straight towards the Earth at
2: Repeat the previous problem with the ship heading directly away from the Earth.
3: If a spaceship is approaching the Earth at
4: (a) Suppose the speed of light were only
5: If a galaxy moving away from the Earth has a speed of
6: A space probe speeding towards the nearest star moves at
7: If two spaceships are heading directly towards each other at
8: Two planets are on a collision course, heading directly towards each other at
9: When a missile is shot from one spaceship towards another, it leaves the first at
10: What is the relative velocity of two spaceships if one fires a missile at the other at
11: 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?
12: 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.
13: Prove that for any relative velocity
14: Show that for any relative velocity
15: (a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy
Glossary
- classical velocity addition
- the method of adding velocities when [latex]{v
- relativistic velocity addition
- the method of adding velocities of an object moving at a relativistic speed:
, where is the relative velocity between two observers, is the velocity of an object relative to one observer, and is the velocity relative to the other observer
- relativistic Doppler effects
- a change in wavelength of radiation that 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
where
is the observed wavelength, is the source wavelength, and is the velocity of the source to the observer
Solutions
Check Your Understanding
Problems & Exercises
1: (a)
(b)
3:
5: a)
b) red
c)
7:
9:
13:
15: a)
b)
c)