Length Contraction (AQA A Level Physics)

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

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)

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:

• 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:

• This is the observed length, L as the pencil is moving relative to observer G

  • Moving between different reference frames tell us how the x and x' are related

• We want to find , 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:

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

• Therefore:

• Where:

  • = the length measured by an observer moving relative to the length being measured (m)

  •  = the proper length (m)

• As   > 1, this means that the  

  • 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