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Length Contraction (HL) (HL IB Physics)

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Ashika

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Length Contraction

  • When objects travel close to the speed of light, to an observer moving relative to that object, it looks as if the object has become shorter
    • This is best demonstrated using rulers
  • Observer H, in their rocket moving close to the speed of light, measures the length of their pencil to be 14 cm
  • Observer G, at rest on Earth, would measure (with remarkable eyesight) the length of the pencil to be shorter
    • They will see lengths contracted in the spaceship from their reference frame, e.g. the length may appear to be 10 cm instead of 14 cm
  • However, the same occurs the other way around
  • For observer H on the rocket, it is observer G that is moving relative to them 
    • Therefore, observer H would measure the length of observer G's pencil as shorter i.e observer H, on the rocket, sees lengths contracted on Earth from their reference frame

1-5-8-length-contraction-ib-2025-physics

A stationary observer in their own reference frame views lengths as shorter in the moving reference frame

Length Contraction Equation

  • The length of an object is the difference in the position of its ends
  • Consider the observers G and H measuring the length of a pencil, which is stationary in reference frame S' (for observer H)

1-5-8-length-contraction-equation

Observer G measures the length of the pencil to be different to observer H

  • As observer H is in the moving frame S', they measure the length of the ruler as:

increment x apostrophe space equals space x subscript 2 apostrophe space minus space x subscript 1 apostrophe space equals space L subscript 0

  • This is the proper length, L0 as the pencil is not moving relative to observer H
    • Both the pencil and observer H are, however, moving relative to observer G
  • Observer G needs to measure the length of the pencil by measuring the position of its ends at the same time (just like observer H did)
  • They measure the length of the ruler to be:

increment x space equals space x subscript 2 space minus space x subscript 1 space equals space L

  • This is the observed length, L as the pencil is moving relative to observer G
    • Lorentz transformations tell us how the x and x' are related
  • We want to find increment x, the length measured in the reference frame of the stationary observer on Earth (G), who is moving relative to the observer on the rocket (H)
  • Transforming these distances gives:

x subscript 1 apostrophe space space equals space gamma open parentheses x subscript 1 space minus space v t close parentheses

x subscript 2 apostrophe space space equals space gamma open parentheses x subscript 2 space minus space v t close parentheses

  • These are then substituted into the equation for the proper length, L0:

x subscript 2 apostrophe space minus space x subscript 1 apostrophe space equals space gamma open parentheses x subscript 2 space minus space v t close parentheses space minus space gamma open parentheses x subscript 1 space minus space v t close parentheses space equals space gamma open parentheses x subscript 2 minus space x subscript 1 close parentheses

L subscript 0 space equals space gamma L

  • Therefore:

L space equals fraction numerator space L subscript 0 over denominator gamma end fraction

  • Where:
    • L = the length measured by an observer moving relative to the length being measured (m)
    • L subscript 0 = the proper length (m)
  • As  gamma > 1, this means that the L space less than thin space L subscript 0 
    • In other words, lengths measured from a reference frame moving relative to the object will be measured as shorter than the lengths measured at rest from within their frame of reference
  • Similar to time dilation, length contraction is also due to Einstein's second postulate
    • Both observers G and H must measure the speed of light to be c
    • Since the time for observer H will run slower, according to observer G (i.e. t increases), then for c to stay the same, the length of the object, L must decrease
  • It is important to note that the length has been measured at the same time
    • This length is the difference between the ends of the pencil, with both ends measured at the same time
  • The ruler used in both reference frames is stationary in their own reference frame
    • Otherwise, observer G would see the ruler on observer H's rocket contracting as well and wouldn't measure any difference in length

Worked example

A spacecraft leaves Earth and moves towards a planet.

The spacecraft moves at a speed of 0.75c relative to the Earth. The planet is a distance of 15 ly away according to the observer on Earth.

1-5-8-length-contraction-worked-example-ib-2025-physics

The spacecraft passes a space station that is at rest relative to the Earth. The proper length of the space station is 482 m.

Calculate the length of the space station according to the observer in the spacecraft.

Answer:

Step 1: List the known quantities

  • Speed of the spacecraft, v = 0.75c
  • Proper length of the space station, L0 = 482 m

Step 2: Analyse the situation

  • We are trying to find the length of the space station in the reference frame of the observer in the spacecraft
  • In this observer's reference frame, it is the space station that is moving away from them at 0.75c
  • Therefore, we a measuring a length in the moving reference frame (relative to the spacecraft) - this is the length, L

Step 3: Substitute values into the length contraction equation

L space equals fraction numerator space L subscript 0 over denominator gamma end fraction space equals fraction numerator space L subscript 0 over denominator fraction numerator 1 over denominator square root of 1 space minus space v squared over c squared end root end fraction end fraction

L space equals space fraction numerator 482 space over denominator fraction numerator 1 over denominator square root of 1 space minus space open parentheses 0.75 c close parentheses squared over c squared end root end fraction end fraction space equals space fraction numerator 482 space over denominator fraction numerator 1 over denominator square root of 1 space minus space open parentheses 0.75 close parentheses squared end root end fraction end fraction space equals space 319 space straight m

Step 4: Check whether your answer makes sense

  • As the observer in the spacecraft is stationary, the length of the space station they measure should be shorter than the proper length 
  • As the length recorded from the spacecraft is 319 years, and the proper length is 482 m, this length makes sense

Examiner Tip

You will not be expected to remember this derivation, but it's helpful to know where all the factors have come from. The time dilation equation is given on your data sheet.

The notion of 'proper length' is incredibly important here, as it depends on the reference frame the length is being measured from. 

You will find in some exam questions you can use time dilation or length contraction, you will receive marks for either way.

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    Ashika

    Author: Ashika

    Expertise: Physics Project Lead

    Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.