Special Relativity Examples: 46 Problems with Answers

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

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

Concept Recap

Special relativity is Einstein's theory describing physics at very high speeds, where measurements of time, length, and simultaneity depend on the observer's frame of reference.

At everyday speeds, classical physics works well. At speeds close to light, time and space behave differently from common intuition.

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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: Special Relativity asks whether the system is nuclear, quantum, or relativistic before using an everyday model.

Common stuck point: Students often know a formula related to special relativity but skip the recognition step: Does the situation involve particles, nuclei, photons, or relativistic speeds where everyday mechanics is not enough? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Does the situation involve particles, nuclei, photons, or relativistic speeds where everyday mechanics is not enough?

Worked Examples

Example 1

medium

A rocket of proper length

100\text{ m}

passes Earth at

v = 0.6c

. What length does an Earth observer measure?

Answer

L = 80 m

First step

Compute

\gamma

v = 0.6c

\gamma = 1/\sqrt{1 - 0.36} = 1.25

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Example 2

medium

A starship makes a round-trip to a star

4\text{ ly}

away at

v = 0.8c

(Earth frame). How long does the trip take in Earth's frame, and how long aboard the ship?

Example 3

medium

A pion has proper lifetime

2.6\times10^{-8}\text{ s}

. In a lab it travels

39\text{ m}

before decaying. Roughly what was its

\gamma

? (Use

v \approx c = 3.00\times10^8\text{ m/s}

Example 4

hard

Two spaceships approach each other, each moving at

0.6c

relative to Earth. Find the speed of one as measured from the other.

Example 5

challenge

An electron is accelerated through a potential difference of

1.0\text{ MV}

. Find

\gamma

and its final speed (rest energy

0.511\text{ MeV}

Practice Problems

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

Example 1

easy

Find the Lorentz factor

\gamma

for an object moving at

v = 0.6c

Example 2

easy

At everyday speeds (far below

c

), is the Lorentz factor close to

1

or much larger?

Example 3

easy

A clock moving at high speed runs slow as seen by a stationary observer. What is this effect called?

Example 4

easy

A moving object appears shortened along its direction of motion. What is this called?

Example 5

easy

Find the rest energy of a

2 \text{ kg}

object using

E = mc^2

(

c = 3\times10^8

Example 6

easy

Should you use relativity formulas for a car moving at

30 \text{ m/s}

Example 7

easy

Find

\gamma

for

v = 0.8c

Example 8

easy

Does 'everything is relative' in physics relativity mean any opinion is as valid as another?

Example 9

medium

A muon's proper lifetime is

2\times10^{-6} \text{ s}

. At

\gamma = 5

, find its lifetime in the lab frame.

Example 10

medium

A rod has proper length

2 \text{ m}

. Moving at

\gamma = 2

, find its contracted length.

Example 11

medium

Find the energy equivalent of

0.5 \text{ kg}

converted entirely to energy (

c = 3\times10^8

Example 12

medium

At what speed (as a fraction of

c

) does

\gamma = 2

Example 13

medium

A spaceship travels

0.6c

. Its clock measures a

4 \text{ year}

trip. How long does an Earth observer measure? (

\gamma = 1.25

)

Example 14

medium

Find

\gamma

for

v = 0.99c

Example 15

medium

A particle's total relativistic energy is

E = \gamma m c^2

. For

\gamma = 3

m = 1\times10^{-27} \text{ kg}

, find

E

(

c = 3\times10^8

Example 16

medium

A clock on a fast ship measures

3 ext{ hours}

for a trip. With

\gamma = 2

, how long does a stationary observer measure?

Example 17

medium

A spaceship of proper length

30 ext{ m}

moves at

\gamma = 1.5

. Find its contracted length.

Example 18

challenge

A muon (

\gamma = 10

, proper lifetime

2.2\times10^{-6} \text{ s}

) moves at nearly

c

. Find the distance it travels in the lab frame (

c = 3\times10^8

Example 19

challenge

Find the kinetic energy of a particle with

\gamma = 2

m = 9.1\times10^{-31} \text{ kg}

using

KE = (\gamma - 1)mc^2

(

c = 3\times10^8

Example 20

challenge

Two events are simultaneous in one frame but not in another moving frame. What principle does this illustrate?

Example 21

easy

What is the Lorentz factor

\gamma

when

v = 0

Example 22

easy

Find the Lorentz factor for

v = 0.5c

Example 23

easy

An electron has mass

m = 9.11 \times 10^{-31}\text{ kg}

. Find its rest energy in joules. Use

c = 3.00 \times 10^8\text{ m/s}

Example 24

easy

Find the Lorentz factor for

v = 0.95c

Example 25

easy

A spaceship moves at

v = 0.6c

. Its onboard clock ticks

1\text{ s}

per tick. How long is one tick as measured on Earth? (

\gamma = 1.25

)

Example 26

medium

A particle has rest mass

m = 1.67\times10^{-27}\text{ kg}

(a proton). Find its total energy at

\gamma = 4

. Use

c = 3.00\times10^8\text{ m/s}

Example 27

medium

At what speed (as a fraction of

c

) does the Lorentz factor equal

\gamma = 1.25

Example 28

medium

A muon's proper lifetime is

2.2\times10^{-6}\text{ s}

. At

v = 0.99c

(

\gamma \approx 7.09

), find its lifetime in the lab frame.

Example 29

medium

Find the Lorentz factor for

v = 0.9c

Example 30

medium

Find the kinetic energy of an electron (

m = 9.11\times10^{-31}\text{ kg}

) at

\gamma = 3

. Use

c = 3.00\times10^8\text{ m/s}

Example 31

medium

A spaceship of proper length

60\text{ m}

is measured to be

30\text{ m}

long by an Earth observer. Find

\gamma

Example 32

medium

Find the relativistic momentum of a proton (

m = 1.67\times10^{-27}\text{ kg}

) at

v = 0.6c

. Use

c = 3.00\times10^8\text{ m/s}

Example 33

medium

Convert

0.511\text{ MeV}

to joules. (

1\text{ eV} = 1.6\times10^{-19}\text{ J}

Example 34

hard

A particle is accelerated until its total energy is

5

times its rest energy. Find its speed as a fraction of

c

Example 35

hard

A spaceship moves at

0.8c

relative to Earth and launches a probe forward at

0.5c

relative to itself. Find the probe's speed relative to Earth.

Example 36

hard

An electron has total energy

E = 1.022\text{ MeV}

. Find its kinetic energy in

\text{MeV}

Example 37

hard

A particle has rest energy

938\text{ MeV}

(a proton) and kinetic energy

2814\text{ MeV}

. Find

\gamma

Example 38

hard

A photon has energy

E

. Find its momentum.

Example 39

hard

A clock on a satellite runs at

\gamma = 1 + 5\times10^{-11}

slower than ground clocks. Roughly how many extra nanoseconds does the satellite clock lose per day relative to ground (consider only this

\gamma

effect)?

Example 40

challenge

A particle has momentum

p = 3mc

. Find its total energy in units of

mc^2

Example 41

challenge

In frame

S

, two events at

x_1 = 0

and

x_2 = 600\text{ m}

are simultaneous. In frame

S'

moving at

v = 0.6c

along

+x

, what is the time difference

\Delta t'

between the events? Use

c = 3\times10^8\text{ m/s}

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Related Concepts

Speed of Light Reference Frame

Background Knowledge

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

speed of lightreference frame