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Theme A — Space, Time and Motion

A.5.8 Length Contraction — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 60 of 63 1 quiz

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A.5.8 Length Contraction — IB PHYSICS HL notes: State and use the length contraction equation and relate it to time dilation.To an observer moving relative to an object near c, the object looks shorter.A 14 cm pencil on a rocket may be measured as 10 cm by an Earth observer.The e…

  • The effect is symmetric
  • The rulers used are stationary in their own frames
  • It is linked to time dilation
  • Sense check
  • Exam questions may be solved with time dilation or length contraction
  • Proper length

Part 1 of 6Length contraction

To an observer moving relative to an object near c, the object looks shorter.
A 14 cm pencil on a rocket may be measured as 10 cm by an Earth observer.
The effect is symmetric
  • The effect is symmetric each observer measures the other's lengths as contracted.
Only the length along the direction of motion contracts.

Part 2 of 6Length contraction equation

L
L0/γ
  • L is the length measured by an observer moving relative to the object (m).
  • L0 is the proper length, measured in the object's rest frame (m).
Since γ is greater than 1, L is less than L0.
The equation is on the data sheet.

Part 3 of 6Derivation from Lorentz transformations

1
Observer H in S' measures the pencil at rest: Δx' = x2' − x1' = L0.
2
Observer G in S measures both ends at the same time: Δx = x2 − x1 = L.
3
Transform the ends: x1' = γ(x1 − vt) and x2' = γ(x2 − vt).
4
Subtract: L0 = γ(x2 − x1) = γL, so L = L0/γ.
  • The derivation is not required for the exam.

Part 4 of 6Conditions for measuring length

1
The ends must be measured at the same time in the measuring frame.
2The rulers used are stationary in their own frames
  • The rulers used are stationary in their own frames a moving ruler would contract as well.
3
Length contraction is a direct consequence of the second postulate.
4It is linked to time dilation
  • It is linked to time dilation both observers must agree on the relative speed v.
  • If clocks in the moving frame run slow by γ, lengths along the motion contract by the same factor.

Part 5 of 6Using length contraction

Using length contraction
Identify the proper length — the length in the frame where the object or distance is at rest.
Calculate γ.
L = L0/γ gives the length seen by the moving observer.
Using length contraction
A spacecraft passing a space station of proper length 482 m sees a shorter length.
Sense check
  • Sense check the measured length must be smaller than the proper length.
Exam questions may be solved with time dilation or length contraction
  • Exam questions may be solved with time dilation or length contraction either method earns marks.

Part 6 of 6More from the specification

More from the specification
Atmospheric path length contraction for muons
  • Atmospheric path length contraction for muons the proper distance between atmospheric creation event and ground detector contracts dramatically in the rest frame of muons.
Proper length definition
  • Proper length definition distance between two spatial points measured by an observer stationary relative to the observed object.
Lorentz length contraction
  • Lorentz length contraction moving objects travelling at relativistic velocity appear shortened along their direction of motion relative to stationary observers.
Galilean versus relativistic frames
  • Galilean relativity assumes absolute space, whereas Lorentz transformations account for the constant invariant speed of light.
Inertial Reference Frames and Relativity
  • Einstein's special theory of relativity applies to inertial reference frames, postulating that the speed of light c is constant for all observers.
    • (frames moving at constant velocity)
Lorentz Transformation Factor (γ)
  • Relativistic phenomena scale with the Lorentz factor γ = 1 / √(1 - v²/c²);
  • as relative velocity v approaches c, γ increases from 1 toward infinity.
Relativistic Length Contraction
  • The spatial length of an object measured by an observer moving at relative velocity v is contracted along the direction of motion according to L = L₀ / γ,
  • where L₀ is proper length.
Time Dilation for Moving Observers
  • A moving clock runs slower relative to a stationary observer by the time dilation equation Δt = γ Δt₀, resolving Galilean transformation discrepancies at near-light speeds.
Reference frame
  • Coordinate system used by an observer.
Velocity addition
  • U = u' + v.
Constant velocity
  • Zero acceleration.
Galilean relativity
  • Newton's laws take the same form in all inertial frames.
Space-time interval
  • Invariant quantity (cΔt)2 − (Δx)2.
Velocity
  • Rate of change of displacement.
Gradient
  • Slope of a graph line.

Key terms

TermMeaning
Length contractionShortening of lengths measured along the direction of relative motion.
Proper lengthLength measured in the rest frame of the object.
Lorentz factorγ = 1/√(1 − v2/c2.
Rest frameFrame in which the object is at rest.
Observed lengthLength measured by an observer moving relative to the object.
Direction of motionAxis along which contraction occurs.

Avoid these mistakes

1
Applying contraction to dimensions perpendicular to the motion.
2
Using L = γL0 — that gives a longer length.
3
Using a moving ruler to define proper length.
4
Taking the proper length as the one measured by the stationary observer.
5
Measuring the two ends at different times.

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