HSC Exam Practice

Sample response. Proper length (L₀) is the length of an object measured in the reference frame in which the object is at rest. It is the longest measured length for that object. Contracted length (L) is the length measured by an observer in a frame in which the object is moving. The contracted length is always shorter than or equal to the proper length (L ≤ L₀).

Marking notes. 1 mark for a correct definition of proper length, explicitly stating the rest frame; 1 mark for a correct definition of contracted length, explicitly stating the moving frame; 1 mark for stating which observer measures each (rest-frame observer measures L₀; observer in relative motion measures L).

Sample response. The formula is L = L₀/γ, where L is the contracted length (m), L₀ is the proper length (m), and γ = 1/√(1−v²/c²) is the Lorentz factor (dimensionless; γ ≥ 1). Only the dimension parallel to the direction of motion contracts. Dimensions perpendicular to the direction of motion are unchanged.

Marking notes. 1 mark for the formula L = L₀/γ; 1 mark for defining L, L₀, and γ (accept partial credit for defining two of three); 1 mark for stating that only the parallel dimension contracts; 1 mark for stating that perpendicular dimensions are unchanged.

Sample response. γ = 1/√(1−v²/c²) = 1/√(1−0.36) = 1/√0.64 = 1/0.8 = 1.25. L = L₀/γ = 250/1.25 = 200 m.

Marking notes. 1 mark for correct calculation of γ = 1.25; 1 mark for correct substitution L = 250/1.25; 1 mark for correct answer 200 m with unit.

Sample response. The space-station observer measures the contracted length L = 200 m. The rocket’s pilot measures the proper length L₀ = 250 m. The pilot’s measurement is the proper length because the pilot is at rest relative to the rocket — both the pilot and the rocket are in the same rest frame. Proper length is always measured in the frame where the object is stationary; all frames in relative motion measure a shorter contracted length.

Marking notes. 1 mark for stating the space-station observer measures 200 m (contracted); 1 mark for stating the pilot measures 250 m (proper length); 1 mark for correctly explaining why the pilot’s measurement is proper length (pilot is in the rocket’s rest frame).

Sample response. The formula L = γL₀ predicts that a moving object is measured to be longer than its proper length, because γ ≥ 1 means γL₀ ≥ L₀. This is the wrong direction: physical length contraction means a moving object appears shorter, not longer. The correct formula L = L₀/γ gives L ≤ L₀ because dividing by γ ≥ 1 makes the measured length shorter, consistent with the observed effect.

Marking notes. 1 mark for identifying the error: L = γL₀ gives L ≥ L₀ (longer, not shorter); 1 mark for stating that the physically correct result is L ≤ L₀; 1 mark for explaining why dividing by γ achieves the correct direction (L = L₀/γ).

Sample response. Cosmic-ray muons are created ~15 km above Earth’s surface and travel at ~0.995c. Without relativistic effects, the classical calculation gives: distance in one proper half-life (2.2 µs) = 0.995 × 3×10⁸ × 2.2×10⁻⁶ ≈ 660 m, so only ~2% of a half-life is needed to travel 15 km classically (about 23 half-lives) — essentially no muons should survive to sea level. Yet muons are detected at sea level at a significant flux (~1 per cm² per min). Length contraction (in the muon’s frame) accounts for this: the atmosphere appears contracted from 15 km to ~1.5 km, which the muon can traverse in about 2 half-lives with ~25% survival.

Marking notes. 1 mark for stating that classical physics predicts essentially no muons reach sea level (very few half-lives to reach the surface); 1 mark for stating that a significant muon flux is detected at sea level; 1 mark for correctly explaining the length-contraction mechanism (atmosphere contracted in muon’s frame) that accounts for the observation.

Part (a) — pole length in barn frame (2 marks). γ = 1.667. L = L₀/γ = 20.0/1.667 = 12.0 m. The contracted pole is exactly 12.0 m, equal to the barn’s proper length. [1 mark correct formula and substitution; 1 mark correct answer 12.0 m with unit.]

Part (b) — barn length in pole frame (2 marks). L_barn = 12.0/1.667 = 7.2 m. [1 mark formula; 1 mark correct answer 7.2 m with unit.]

Part (c) — resolution of apparent paradox (3 marks). In the barn frame, the contracted pole (12.0 m) exactly fits inside the 12.0 m barn, so it is physically possible for both doors to be closed simultaneously at the instant the pole is inside [1]. In the pole’s rest frame, the barn (7.2 m) cannot contain the 20.0 m pole, so in that frame the two door-closing events are never simultaneous while the full pole is enclosed [1]. The relativistic principle that resolves the paradox is the relativity of simultaneity: two spatially separated events (front door closes; back door closes) that are simultaneous in one inertial frame (barn frame) are not simultaneous in another inertial frame moving relative to the first (pole frame). Both observers are self-consistent; there is no physical contradiction because “both doors closed at the same time” is a frame-dependent statement, not an absolute event [1].

Sample response. The claim that length contraction and time dilation are “two separate relativistic effects” is misleading. While they appear to be distinct — one describing spatial measurements and the other temporal measurements — they are inseparable consequences of the same underlying Lorentz transformation of spacetime coordinates. Consider the muon example: muons created at ~15 km altitude travel at 0.995c with γ ≈ 10. From Earth’s frame (time dilation): the muon’s proper half-life of 2.2 µs is dilated to γ × 2.2 = 22 µs. Distance travelled in one dilated half-life = 0.995c × 22×10⁻⁶ ≈ 6.6 km. The muon is created at ~15 km, so it takes about 15/6.6 ≈ 2.3 half-lives to reach the surface: survival fraction ≈ (0.5)^2.3 ≈ 20%. From the muon’s frame (length contraction): the atmosphere is contracted from 15 km to L = 15/10 = 1.5 km. The muon travels 1.5 km in its own proper half-life: time = 1.5×10³/(0.995×3×10⁸) ≈ 5 µs ≈ 2.3 proper half-lives. Survival fraction again ≈ 20%. Both calculations yield the same quantitative prediction. This is not a coincidence: in the Lorentz transformation, the spatial contraction and temporal dilation arise simultaneously from the same set of equations; you cannot apply one without the other. If length were not contracted by the same factor as time is dilated, the speed of light would not be invariant between frames — a direct contradiction of Einstein’s second postulate. Therefore, viewing them as “separate effects” is technically incorrect as a matter of physics: they are two manifestations of a single four-dimensional spacetime geometry. The appropriate description is that both effects are frame-dependent projections of the invariant spacetime interval ds² = c²dt² − dx². Observers in different inertial frames “slice” this interval differently between its temporal and spatial components, producing time dilation in one direction and length contraction in the perpendicular (spatial) direction. In summary, the claim is an oversimplification: length contraction and time dilation are related by necessity, not by coincidence, and a full description of relativistic kinematics requires both.

Marking criteria (9 marks): 1 = correctly states that length contraction and time dilation are related (not fully independent), with a brief reason. 1 = time-dilation calculation using the muon example: correct dilated half-life (γ × 2.2 µs = 22 µs) or equivalent. 1 = length-contraction calculation using the muon example: L = 15.0/10 = 1.5 km or equivalent. 1 = both calculations yield the same predicted survival fraction or distance, demonstrating they are consistent. 1 = explains that the Lorentz transformation produces both effects simultaneously from the same mathematical framework. 1 = links the invariance of the speed of light to the necessity of both effects occurring together. 1 = identifies the spacetime interval as the invariant underlying both effects, or equivalently discusses the four-dimensional geometry. 1 = reaches an explicit evaluative judgement: calling them “separate” is an oversimplification; they are aspects of the same spacetime geometry. 1 = precise technical language throughout (γ, Lorentz factor, proper/contracted length, proper/dilated time, inertial frame, spacetime interval, Lorentz transformation, postulate).