Physics • Year 12 • Module 7 • Lesson 13
Length Contraction
Build HSC Band 5–6 extended-response technique on analysing length contraction scenarios, resolving apparent paradoxes, and evaluating the relationship between length contraction and time dilation.
1. The barn-and-ladder paradox — extended analysis (Band 5–6)
8 marks Band 5–6
Scenario. A barn has proper length 10.0 m. A ladder has proper length 15.0 m. The ladder moves at 0.80c (Lorentz factor γ = 5/3 ≈ 1.667) directly along the barn’s axis. The barn has two doors, one at each end, that can be opened and closed by sensors. A student in the barn claims: “When the ladder’s contracted length equals 9.0 m, I can close both doors simultaneously and briefly trap the ladder inside the 10.0 m barn.” A student riding on the ladder claims: “From my frame, the barn is contracted to 6.0 m. My 15.0 m ladder cannot possibly fit, so both doors cannot ever be simultaneously closed while the ladder is inside.”
Q1. Analyse and evaluate both students’ claims to determine whether they are each self-consistent, and explain why there is no genuine physical contradiction. In your response you must:
- Calculate the contracted length of the ladder in the barn’s frame and verify that the barn student’s claim is numerically correct.
- Calculate the contracted length of the barn in the ladder’s frame and verify that the ladder student’s claim is numerically correct.
- Identify and explain the relativistic concept that resolves the apparent paradox — why both observers are correct even though their claims seem contradictory.
- Describe what each observer actually experiences regarding the two door-closing events: are the events simultaneous in the barn frame? In the ladder frame?
- State a general principle about physical outcomes (absolute events) in special relativity that prevents genuine contradictions.
2. Experimental design — testing relativistic length contraction in the laboratory (Band 5–6)
7 marks Band 5–6
Research question. A physics teacher claims that it is impossible to observe length contraction directly in a school laboratory because no object can be accelerated to an appreciable fraction of c. A student counters by suggesting that relativistic particles (such as the muons in a cloud chamber or electrons accelerated in a cathode-ray tube) could be used to test the effect indirectly. Design a scientific investigation to test whether relativistic length contraction is consistent with experimental observations, using accessible particle-physics data (muon flux measurements at sea level vs altitude, or published electron scattering data).
Constraints: You may use secondary data (published values of muon flux, half-life, and altitude). Provide quantitative reasoning in your design. Your investigation must address at least one potential systematic error.
Q2. Design the investigation and present it in the format below.
- State your research question in the form: “Does length contraction [specific prediction] agree with [measured quantity]?”
- State a testable hypothesis including the independent and dependent variables and the specific quantitative prediction (using L = L0/γ).
- Describe the procedure: what data you will use, how you will calculate the predicted contracted length, and how you will compare it to the measured outcome.
- Identify one systematic error and explain how it might affect the conclusion.
- State what result would falsify the hypothesis.
- Discuss what it would mean if the muon flux at sea level is much higher than classical (non-relativistic) physics predicts.
Q1 — Sample Band 6 response (8 marks), annotated
Ladder in barn frame (calculation): γ = 1.667 (given). Contracted ladder length = L0/γ = 15.0/1.667 = 9.0 m. Since 9.0 m < 10.0 m (barn proper length), the barn student’s claim is numerically correct: the contracted ladder fits inside the barn. [1 — correct calculation with formula and conclusion]
Barn in ladder frame (calculation): Contracted barn length = 10.0/1.667 = 6.0 m. Since 6.0 m < 15.0 m (ladder proper length), the ladder student’s claim is also numerically correct: the barn cannot contain the full ladder in this frame. [1 — correct calculation with formula and conclusion]
Relativistic concept resolving the paradox: The apparent contradiction arises because “simultaneously enclosing the ladder” requires two spatially separated events — closing the front door and closing the back door — to happen at the same time. In special relativity, simultaneity is frame-dependent (relativity of simultaneity): two events that are simultaneous in one inertial frame are not simultaneous in a frame moving relative to the first. There is no privileged frame in which “really simultaneous” is defined. Both observers are self-consistent; they simply disagree about whether two spatially separated events are simultaneous. [1 — names “relativity of simultaneity”; 1 — explains that two spatially separated events can be simultaneous in one frame but not in another]
What each observer experiences: In the barn’s rest frame, sensors trigger both doors to close at the same time: the contracted 9.0 m ladder is entirely inside the 10.0 m barn at the instant both doors are closed. In the ladder’s rest frame, the front door of the barn closes first (when the front of the ladder reaches it); the back door only closes later (when the back of the ladder reaches it). At no single instant in the ladder frame are both doors simultaneously closed while the entire 15.0 m ladder is inside the 6.0 m barn. [1 — barn frame describes simultaneous closing; 1 — ladder frame describes non-simultaneous closing with front door closing first]
General principle: Physical outcomes that are absolute events (e.g. whether a specific detector fires, whether a photograph is taken of the ladder inside the barn) are agreed upon by all observers. The statement “was the ladder simultaneously enclosed?” is frame-dependent (not an absolute event), so there is no paradox. Absolute events (colocal in spacetime) cannot change between frames; only frame-dependent statements (simultaneity of spatially separated events) can differ. [1 — states the principle clearly; 1 — applies it to distinguish frame-dependent simultaneity from absolute physical outcomes]
Marking criteria (8 marks): 1 = correct numerical ladder length in barn frame (9.0 m) with formula; 1 = correct numerical barn length in ladder frame (6.0 m) with formula; 1 = names “relativity of simultaneity” as the resolving principle; 1 = explains that simultaneous events in one frame need not be simultaneous in another; 1 = describes barn frame: both doors close simultaneously with ladder inside; 1 = describes ladder frame: front door closes before back door; no simultaneous enclosure; 1 = states the general principle about absolute vs frame-dependent events; 1 = uses precise technical language throughout (Lorentz factor, inertial frame, simultaneity, contracted length, rest frame).
Q2 — Sample Band 6 response (7 marks), annotated
Research question: Does the length-contraction prediction (L = L0/γ ≈ 1.5 km for the atmosphere in the muon’s frame) agree with the observed sea-level muon flux that would be impossible to explain using classical (non-relativistic) physics? [1 — specific quantitative research question]
Hypothesis: If length contraction is real, then the atmosphere appears contracted from 15.0 km to approximately 1.5 km in the muon’s frame (γ ≈ 10.0 at v = 0.995c). In that frame, the muon need only travel 1.5 km in its proper lifetime of 2.2 µs. The independent variable is whether we apply relativistic length contraction (or equivalently, time dilation). The dependent variable is the predicted survival fraction of muons at sea level. The quantitative prediction: ≈ 1.5 km / (0.995c × 2.2 µs) ≈ 2.3 half-lives, giving a survival fraction of (0.5)2.3 ≈ 0.20 (20%). Classical prediction: 15 km / (0.995 × 3×108 m/s × 2.2×10−6 s) ≈ 23 half-lives, giving a survival fraction of (0.5)23 ≈ 10−7 (essentially zero). [1 — quantitative hypothesis with IV and DV; 1 — correct numerical prediction]
Procedure: (1) Use published values: muon production altitude ~15 km, muon proper half-life 2.197 µs, sea-level muon flux (measured by CERN and others: ~1 muon cm−2 min−1 at sea level). (2) Calculate the predicted flux using non-relativistic decay: surviving fraction ≈ 10−7 of the upper-atmosphere production rate → predicted sea-level flux essentially zero. (3) Calculate the relativistic prediction: surviving fraction ≈ 20%, consistent with the observed ~1 muon cm−2 min−1. (4) Compare measured sea-level flux to both predictions. [1 — procedure with at least three steps and a clear comparison method]
Systematic error: The muon production altitude is not a sharp boundary at exactly 15 km; muons are created across a range of altitudes (10–20 km depending on primary cosmic-ray energy). This introduces uncertainty in L0 and hence in the predicted survival fraction. To reduce this error, use a mean effective production altitude from published CORSIKA simulation data, and calculate a range of predicted fractions. [1 — one valid systematic error identified and its effect explained]
Falsification: If the measured sea-level muon flux matched the classical non-relativistic prediction (essentially zero per cm² per minute, as expected from ~10−7 survival), the hypothesis of length contraction would be falsified — muons could not reach sea level without the relativistic effect. [1 — correct falsification criterion]
Significance of high sea-level muon flux: The measured sea-level flux is many orders of magnitude higher than classical physics predicts. This is direct empirical evidence that relativistic effects (length contraction in the muon’s frame, equivalent to time dilation in Earth’s frame) are operating: the muon’s effective range is increased by a factor of ~γ = 10 relative to the classical expectation. The data strongly support special relativity. [1 — explains the significance in terms of the relativistic prediction being confirmed and classical physics being ruled out]
Marking criteria (7 marks): 1 = specific quantitative research question; 1 = hypothesis with IV and DV; 1 = correct quantitative prediction (relativistic vs classical); 1 = procedure with data source, calculation steps, and comparison; 1 = one valid systematic error with effect on conclusion; 1 = correct falsification criterion; 1 = explains significance of measured flux ruling out classical physics.