Redshift Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Redshift.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Redshift is the increase in the observed wavelength of light, usually because a light source is moving away from the observer or because space itself is expanding and stretching the light as it travels.

When light is stretched, it shifts toward the red end of the spectrum.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Redshift starts by following rays or wavefronts through boundaries, materials, and image locations.

Common stuck point: Students often know a formula related to redshift but skip the recognition step: Am I tracking how light travels through space or materials, including boundary rules and image location when needed? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I tracking how light travels through space or materials, including boundary rules and image location when needed?

Worked Examples

Example 1

medium

A line at rest

434\text{ nm}

(Balmer H

\gamma

) is observed at

521\text{ nm}

. Find

z

and estimate

v

(small-

z

approximation).

Answer

z \approx 0.200, v \approx 6.0 \times 10^{7} m/s

First step

z = (521 - 434)/434 = 87/434 \approx 0.200

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Example 2

medium

A galaxy line at rest

589\text{ nm}

is observed at

618.45\text{ nm}

. Find

z

and the recession speed.

Example 3

hard

A galaxy spectrum shows lines that are systematically 4.0% longer in wavelength than the lab values. Find

z

, the recession speed (small-

z

), and the distance (

H_0 = 70\text{ km/s/Mpc}

Example 4

hard

Two emission lines, originally at

400\text{ nm}

and

600\text{ nm}

, are seen at

440\text{ nm}

and

660\text{ nm}

. Verify both yield the same

z

and state

z

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A galaxy's light shows increased wavelength compared to its rest wavelength. What is this called?

Example 2

easy

Redshift usually indicates a light source is moving how relative to the observer?

Example 3

easy

If a source moves toward the observer, the light shifts toward which end of the spectrum?

Example 4

easy

Does redshift mean the light has become dimmer?

Example 5

easy

Define the redshift parameter

z

in terms of observed and rest wavelengths.

Example 6

easy

A red-colored apple is illuminated by white light. Is this an example of redshift?

Example 7

easy

Cosmological redshift is caused by what large-scale phenomenon?

Example 8

easy

A spectral line has rest wavelength 500 nm and is observed at 500 nm. What is the redshift

z

Example 9

medium

A spectral line has rest wavelength 600 nm and is observed at 660 nm. Find the redshift

z

Example 10

medium

For small redshift,

z \approx v/c

. A galaxy has

z = 0.02

. Estimate its recession speed.

Example 11

medium

A line at rest wavelength 486 nm is observed at 510 nm. Find

z

, then the recession speed using

v\approx zc

Example 12

medium

A galaxy recedes at

v = 3.0 \times 10^6 \text{ m/s}

. Estimate its redshift

z

(small-

z

Example 13

medium

A spectral line is redshifted with

z = 0.5

. Find the observed wavelength if the rest wavelength is 400 nm.

Example 14

medium

Two galaxies have

z = 0.01

and

z = 0.04

. By Hubble's law, which is farther, and by what factor (assuming linear

v

-distance)?

Example 15

challenge

A quasar shows

z = 2.0

. A line emitted at 121.6 nm (ultraviolet) is observed at what wavelength, and is it now visible?

Example 16

challenge

A line at rest 656 nm is observed at 1312 nm. Find

z

and the observed wavelength's spectral region.

Example 17

challenge

Using Hubble's law

v = H_0 d

with

H_0 = 70 \text{ km/s/Mpc}

, a galaxy has

z = 0.01

. Estimate its distance in Mpc (use

v \approx zc

c = 3.0\times10^5 \text{ km/s}

Example 18

medium

A spectral line shifts from rest 400 nm to observed 440 nm. Find the redshift

z

Example 19

medium

A galaxy has redshift

z = 0.05

. Estimate its recession speed using

v\approx zc

Example 20

medium

A line at rest 500 nm has redshift

z = 0.20

. Find the observed wavelength.

Example 21

easy

A spectral line has rest wavelength

450\text{ nm}

and is observed at

459\text{ nm}

. Find the redshift

z

Example 22

easy

A line at rest

500\text{ nm}

is observed at

505\text{ nm}

. Find

z

Example 23

easy

A line at rest

620\text{ nm}

is observed at

651\text{ nm}

. Find

z

Example 24

easy

A galaxy has

z = 0.03

. Estimate its recession speed using

v \approx zc

(with

c = 3.0\times 10^8\text{ m/s}

Example 25

medium

A galaxy recedes at

v = 1.5 \times 10^7\text{ m/s}

. Estimate its redshift

z

(small-

z

Example 26

medium

A line at rest

656\text{ nm}

is observed at

689\text{ nm}

. Find

z

(3 sig figs).

Example 27

medium

A line has rest wavelength

550\text{ nm}

and the source recedes at

v = 6.0\times 10^6\text{ m/s}

. Find the observed wavelength (small-

z

Example 28

medium

A line at rest

720\text{ nm}

is observed at

792\text{ nm}

. Find

z

and the observed wavelength change.

Example 29

medium

Using

v = H_0 d

with

H_0 = 70\text{ km/s/Mpc}

and

c = 3.0\times 10^5\text{ km/s}

, find the distance of a galaxy with

z = 0.02

Example 30

medium

Two galaxies have

z_1 = 0.005

and

z_2 = 0.015

. By Hubble's law, by what factor is galaxy 2 farther than galaxy 1?

Example 31

medium

If a line at rest

410\text{ nm}

is observed at

451\text{ nm}

, find

z

Example 32

medium

A line at rest

700\text{ nm}

has redshift

z = 0.15

. Find the observed wavelength.

Example 33

medium

A line at rest

300\text{ nm}

(UV) is observed at

360\text{ nm}

. Find

z

and identify the spectral region of the observed line.

Example 34

medium

A line at rest

500\text{ nm}

shows recession at

9\times 10^6\text{ m/s}

. Find the observed wavelength.

Example 35

hard

A quasar shows

z = 3.0

. A line emitted at

121.6\text{ nm}

(Lyman-

\alpha

) is observed at what wavelength?

Example 36

hard

Using

v = H_0 d

with

H_0 = 70\text{ km/s/Mpc}

, a galaxy is at

200\text{ Mpc}

. Estimate its redshift

z

Example 37

hard

A spectral line at rest

500\text{ nm}

is observed at

1000\text{ nm}

. Find

z

Example 38

hard

A spectrum shows

\lambda_{obs} = 3\lambda_{rest}

. Find

z

Example 39

hard

A line at rest

122\text{ nm}

is observed at

976\text{ nm}

. Find

z

Example 40

challenge

Using the relativistic Doppler formula

1+z = \sqrt{\frac{1+\beta}{1-\beta}}

with

\beta = v/c

, find

z

for

\beta = 0.5

(3 sig figs).

Example 41

challenge

A photon with rest frequency

f_{rest} = 5.0\times 10^{14}\text{ Hz}

is observed from a source at

z = 0.25

. Find the observed frequency.

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Related Concepts

Doppler Effect Visible Light

Background Knowledge

These ideas may be useful before you work through the harder examples.

doppler effectvisible light