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
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
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
| Term | Meaning |
|---|---|
| Length contraction | Shortening of lengths measured along the direction of relative motion. |
| Proper length | Length measured in the rest frame of the object. |
| Lorentz factor | γ = 1/√(1 − v2/c2. |
| Rest frame | Frame in which the object is at rest. |
| Observed length | Length measured by an observer moving relative to the object. |
| Direction of motion | Axis 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.
Marked practiceQuizzes for this lesson
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