Select your language

Suggested languages for you:
Log In Start studying!
StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

All-in-one learning app

  • Flashcards
  • NotesNotes
  • ExplanationsExplanations
  • Study Planner
  • Textbook solutions
Start studying

Length Contraction

Save Save
Print Print
Edit Edit
Sign up to use all features for free. Sign up now
Length Contraction

Imagine you are on a road trip with a friend. You say that you have 20km left to go, while your friend says that you have 30km left. You may disagree, but the disagreement can be resolved by measuring the distance ahead. At least, that is the case when you are travelling at everyday speeds.

However, when travelling at relativistic speeds, which are speeds close to the speed of light, the disagreement between two observers at different reference frames may be significant.

Length Contraction. Distance in length contraction. StudySmarterFigure 1. Distances may be described differently by various people, but at relativistic speeds, the distances are truly different. Source: John Christian Fjellestad, Flickr (CC BY 2.0).

What is length contraction?

We speak of length contraction when the length of an object travelling at a certain speed with respect to a frame of reference is measured to be shorter than its proper length. Proper length (L0) is the distance between two points observed by an observer who is at rest relative to both points.

Despite the fact that clocks measure different elapsed periods for the same procedure, relative speed, which is distance divided by elapsed time, is the same. This means that distance is also affected by the relative motion of the observer. These two affected quantities cancel each other out, leaving the speed to remain constant. For relative speed to be the same for two observers who see different times, they must also measure different distances.

Length contraction is the phenomenon that states that when an observer is moving at a speed close to the speed of light, distances obtained by various observers are not the same.

How do you calculate length contraction?

Having discussed what we mean by length contraction and proper length, lets look at an example to explore how to calculate length contraction.

Lets say a spaceship is moving at a velocity v that is close to the speed of light. An observer A on the earth and an observer B in the spaceship will observe different lengths for the distance covered by the spaceship.

Length Contraction. StudySmarterFigure 2. Length contraction. Source: Manuel R. Camacho, StudySmarter.

We know that the velocity of the spaceship is the same for all observers. If we calculate the velocity v relative to the earth-bound observer A, we get:

Here, L0 is the proper length observed by the earth-bound observer A, while Δt is the time relative to the earth-bound observer A.

The velocity relative to the moving observer B is:

Here, Δt0 is the proper time observed by the moving observer B, while L is the distance observed by the moving observer B.

The two velocities are the same:

We know from time dilation that t = t0. Entering this into the previous equation, we get:

We also know that:

Inserting y, we get the equation for length contraction as shown below:

Length contraction of an objects length

One of the consequences of length contraction is that if an object is moving at a velocity near the speed of light, its length may be observed to be less than its proper length by an observer who is at rest relative to the motion. Lets consider the following example.

Length Contraction. StudySmarterFigure 3. Length contraction. Source: Oğulcan Tezcan, StudySmarter.

Take a 10cm stick. Its length will no longer seem to be 10cm if it passes you at a speed near the speed of light.

The length of the stick while at rest is referred to as its proper length. When the stick is moving near the speed of light, the length measured will always be less than the proper length. When the sticks speed is equal to the speed of light, the stick should, in theory, have no length.

What is an example of length contraction?

A great example of length contraction is when an object is travelling through space, as in the following example.

Lets imagine an observer is travelling from the blue planet to the red one and travelling at the speed of y=30.00. The distance between the two planets is 4,000 light-years as measured by an earth-bound observer. What is the distance relative to the observer on the spaceship in measured kilometres?

Length Contraction. StudySmarterFigure 4. Length contraction. Source: Oğulcan Tezcan, StudySmarter.

If 4,000 light-years is the distance measured by the earth-bound observer, then this is the proper length L0. As we said, the relationship between proper length L0 and the length observed by the moving observer is:

Adding the known variables L0 and y gives us:

1 light-year equals 9.46 ⋅ 10^12 kilometres.

Hence,.

Length Contraction - Key takeaways

  • Length contraction is the phenomenon that occurs when the length of an item travelling at a certain speed is measured to be shorter than its proper length.
  • Proper length (L0) is the distance between two points observed by an observer who is at rest relative to both points.
  • If an object is moving at a velocity near the speed of light, its length will be observed to be less than its proper length by an observer who is at rest relative to the moving object.

Frequently Asked Questions about Length Contraction

Length contraction is the phenomenon that occurs when the length of an item traveling at a certain speed is measured to be shorter than its proper length.


Length contraction is caused by the fact that the speed of light in a vacuum is constant in any frame of reference.

When the length of an item travelling at a certain speed is measured to be less than its proper length, this is known as length contraction. Time dilation is the phenomenon by which time is measured differently for objects travelling through space than for stationary objects.

Final Length Contraction Quiz

Question

Which of the following defines proper length?

Show answer

Answer

The distance between two points observed by an observer who is at rest relative to both points.

Show question

Question

Which of the following defines proper time?

Show answer

Answer

Proper time is the time measured by an observer at rest relative to the event being observed.

Show question

Question

What is the relationship between the measured length of an object travelling at relativistic speeds and proper length?

Show answer

Answer

The measured length will be shorter than the proper length.

Show question

Question

What is meant by relativistic speeds?


Show answer

Answer

Speeds at which relativistic effects appear.

Show question

Question

Can an object’s own length be affected by length contraction? If so, how?

Show answer

Answer

Yes, it can be affected. Its own length can be observed as less than its proper length.

Show question

Question

Consider an 8cm stick moving at the speed of light. In theory, what length will it appear to have to an observer at rest relative to the moving stick?

Show answer

Answer

It will appear to have no length.

Show question

Question

What is 1 light-year in kilometres?

Show answer

Answer

9.46*10^12.

Show question

Question

What is the speed of light in m/s?

Show answer

Answer

3.00*10^8.

Show question

Question

What does Δt0 stand for?

Show answer

Answer

Proper time.

Show question

Question

Which of the following quantities will be the same for all observers when length contraction occurs?

Show answer

Answer

The velocity of the moving object.

Show question

Question

Distance is affected by the relative motion of the observer. True or false?

Show answer

Answer

True.

Show question

Question

When an object is moving near the speed of light, the length measured will always be more than the proper length. True or false?

Show answer

Answer

False.

Show question

More about Length Contraction
60%

of the users don't pass the Length Contraction quiz! Will you pass the quiz?

Start Quiz

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Just Signed up?

Yes
No, I'll do it now

Sign up to highlight and take notes. It’s 100% free.