Physics • Year 12 • Module 7 • Lesson 13
Length Contraction
Lock in the core vocabulary, the length contraction formula, and the key properties of proper and contracted length before tackling harder questions.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from this list: proper length, contracted length, length contraction, Lorentz factor (γ), rest frame, inertial frame, proper time, simultaneity, relativistic, barn-ladder paradox. 10 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The length of an object measured in the frame of reference in which the object is at rest. | |
| 1.2 | The shorter length measured by an observer in a frame in which the object is moving. | |
| 1.3 | The relativistic phenomenon whereby the measured length of an object moving relative to an observer is shorter than its proper length, in the direction of motion only. | |
| 1.4 | The quantity γ = 1/√(1−v²/c²); always greater than or equal to 1. | |
| 1.5 | The reference frame in which the object being measured is stationary. | |
| 1.6 | A reference frame that moves at constant velocity (no acceleration), in which Newton’s first law holds. | |
| 1.7 | The time interval measured by a clock that is present at both events being timed — the shortest possible measured time for that pair of events. | |
| 1.8 | The concept that two spatially separated events that are simultaneous in one inertial frame may not be simultaneous in another frame moving relative to the first. | |
| 1.9 | Describing effects that become significant only at speeds that are an appreciable fraction of the speed of light. | |
| 1.10 | A thought experiment in which a ladder moving at high speed appears to fit inside a shorter barn in the barn’s rest frame, yet appears too long in the ladder’s rest frame — resolved by the relativity of simultaneity. |
2. True or false — with correction
Circle T or F for each statement. If the statement is false, write the corrected version on the line below it. 12 marks (1 T/F + 1 correction each)
2.1 A moving object is measured to be longer than its proper length by a stationary observer. T / F
2.2 The formula for length contraction is L = L0/γ, where L0 is the proper length. T / F
2.3 Length contraction applies equally to all three spatial dimensions of a moving object. T / F
2.4 The proper length is the longest length that can be measured for an object — all other frames measure a length that is shorter or equal. T / F
2.5 Since γ ≥ 1 always, the contracted length L is always less than or equal to the proper length L0. T / F
2.6 Time dilation and length contraction are two independent effects with no physical connection. T / F
3. Fill-in-the-blank paragraph
Use the word bank to complete the passage. Each word is used once. 8 marks (1 per blank)
Word bank:
contracted · direction · gamma · moving · parallel · proper · shorter · unchanged
Length contraction is a relativistic effect in which the measured length of a ___________ object is ___________ than its proper length. The ___________ length, L0, is defined as the length measured in the object’s own rest frame. An observer in relative motion measures the ___________ length L = L0/___________. Contraction occurs only along the ___________ of motion; dimensions ___________ to the direction of motion remain ___________.
4. Function recall
Answer each question in 1–2 sentences using precise terms from the lesson. 8 marks (2 each)
4.1 State the length contraction formula. Define every symbol and state the unit for each quantity.
4.2 Explain why length contraction does not affect the dimensions of a moving object that are perpendicular to its velocity.
4.3 State one real-world observational example of length contraction that provides indirect evidence for the effect. Explain what is observed and which frame undergoes the contraction.
4.4 A student writes L = γL0. Explain the error and write the correct equation.
5. Formula substitution — scaffolded calculations
Follow the steps to complete each calculation. Show all working. 6 marks (2 each)
5.1 A rod has proper length 10.0 m. It moves at 0.6c.
Step 1: Calculate γ. γ = 1/√(1 − v²/c²) = 1/√(1 − __________) = __________
Step 2: Calculate L = L0/γ = __________ / __________ = __________ m
5.2 A spacecraft has a proper length of 400 m. Earth observers measure its length as 200 m. Find γ and hence the spacecraft’s speed.
Step 1: γ = L0/L = __________ / __________ = __________
Step 2: v = c√(1 − 1/γ²) = c√(1 − __________) = __________ m/s (≈ __________ c)
5.3 In the muon scenario, a muon travels from the top of Earth’s atmosphere (proper length 15.0 km) at 0.995c. Calculate the contracted length of the atmosphere in the muon’s rest frame. (γ at 0.995c ≈ 10.0)
L = L0/γ = __________ / __________ = __________ km
6. Build a concept map
Draw labelled arrows between the six terms below to show how they connect. Each arrow must carry a linking phrase (e.g. “causes”, “is divided by”, “is the maximum value of”). Aim for at least 6 labelled arrows. 6 marks (1 per valid labelled arrow)
Supplied terms: proper length (L0) · contracted length (L) · Lorentz factor (γ) · velocity (v) · speed of light (c) · rest frame.
Q1 — Term–definition match
1.1 proper length • 1.2 contracted length • 1.3 length contraction • 1.4 Lorentz factor (γ) • 1.5 rest frame • 1.6 inertial frame • 1.7 proper time • 1.8 simultaneity • 1.9 relativistic • 1.10 barn-ladder paradox.
Q2 — True / false with correction
2.1 False. A moving object is measured to be shorter (not longer) than its proper length. L = L0/γ and since γ ≥ 1, L ≤ L0.
2.2 True. L = L0/γ is the correct formula.
2.3 False. Length contraction affects only the dimension parallel to the direction of motion. Perpendicular dimensions are unchanged.
2.4 True. Proper length L0 is the maximum; all other inertial observers measure L ≤ L0.
2.5 True. Since γ ≥ 1, dividing L0 by γ gives L ≤ L0.
2.6 False. Time dilation and length contraction are two aspects of the same underlying spacetime geometry. You cannot have one without the other; they both follow from the Lorentz transformations.
Q3 — Cloze paragraph
In order: moving / shorter / proper / contracted / gamma / parallel / direction / unchanged.
Q4.1 — Length contraction formula
L = L0/γ, where L is the contracted length measured by the moving observer (m), L0 is the proper length in the object’s rest frame (m), and γ = 1/√(1−v²/c²) is the dimensionless Lorentz factor.
Q4.2 — Why perpendicular dimensions are unchanged
A thought experiment shows that if perpendicular lengths were contracted, a rod moving transversely through a ring would either fit through or not depending on which frame you use — a contradiction. Einstein’s postulates require that transverse measurements agree between frames, so only the dimension parallel to relative motion can contract.
Q4.3 — Real-world observational evidence
Muons created in the upper atmosphere (at ~15 km altitude) travel at ~0.995c and are detected at Earth’s surface. In the muon’s rest frame, the atmosphere is length-contracted to ~1.5 km, which the muon can traverse within its 2.2 µs lifetime. This indirect observation is consistent with the length contraction prediction.
Q4.4 — Common equation error
The student has inverted the formula. L = γL0 would give a contracted length larger than proper length, which is physically wrong. The correct formula is L = L0/γ. Since γ ≥ 1, dividing by γ correctly gives L ≤ L0.
Q5.1 — Rod at 0.6c
γ = 1/√(1−0.36) = 1/√0.64 = 1/0.8 = 1.25. L = 10.0/1.25 = 8.0 m.
Q5.2 — Spacecraft contracted to 200 m
γ = 400/200 = 2.00. v = c√(1−1/4) = c√0.75 = 0.866c ≈ 2.60 × 108 m/s.
Q5.3 — Muon atmosphere
L = 15.0/10.0 = 1.50 km (1500 m). The muon sees the atmosphere compressed to 1.5 km, which it can cross in about 5 µs — much longer than its rest-frame lifetime of 2.2 µs would allow without contraction.
Q6 — Sample concept map
Correct maps should include arrows such as:
- proper length L0 — is measured in → rest frame
- proper length L0 — is divided by → Lorentz factor γ — to give → contracted length L
- velocity v — determines → Lorentz factor γ
- Lorentz factor γ — approaches infinity as v approaches → speed of light c
- contracted length L — is always ≤ → proper length L0
- rest frame — gives maximum length → proper length L0
Award 1 mark per valid labelled arrow (minimum 6, maximum 6 marked).