HSC Exam Practice

Sample response. Proper time (Δt⊂0;) is the time interval between two events measured by a single clock that is present at both events — the clock in the frame where the events occur at the same position. Dilated time (Δt) is the time interval between the same two events measured by an observer in a frame moving relative to that clock. Proper time is always the shorter interval because the Lorentz factor γ ≥ 1, so Δt = γΔt⊂0; ≥ Δt⊂0;.

Marking notes. 1 mark for a correct definition of proper time (single clock present at both events / same-position frame). 1 mark for correct definition of dilated time (moving frame measurement). 1 mark for stating proper time is shorter and linking to γ ≥ 1.

Sample response. The two postulates are: (1) The laws of physics are the same in all inertial frames of reference. (2) The speed of light in a vacuum is constant (c ≈ 3.0 × 10⁸ m s⁻¹) for all inertial observers, independent of the motion of the source or observer. In the light clock, a photon bounces between two mirrors separated by distance d. In the clock’s rest frame, one tick takes Δt⊂0; = 2d/c. When the clock moves at speed v, the photon must travel a longer diagonal path. Since c is the same for all observers (postulate 2), the longer path requires more time: Δt = γΔt⊂0;, so the moving clock ticks slower.

Marking notes. 1 mark for each postulate correctly stated. 1 mark for explaining that the photon travels a longer diagonal path when the clock moves. 1 mark for applying the constancy of c to conclude that the longer path requires longer time, leading to time dilation.

Sample response. Muons are created in the upper atmosphere (~15 km altitude) when cosmic rays strike air molecules. In their own rest frame their mean lifetime is τ⊂0; = 2.2 µs. At 0.99c, classical physics predicts they travel d = 0.99c × 2.2 × 10⁻⁶ ≈ 650 m before decaying, so they should not reach sea level. However, with γ ≈ 7, time dilation extends their lifetime in Earth’s frame to Δt = γτ⊂0; = 7 × 2.2 ≈ 15.4 µs. The distance travelled becomes d = 0.99c × 15.4 × 10⁻⁶ ≈ 4.6 km. In practice, muons at 0.999c have γ ≈ 22 and can traverse the full 15 km. The observed abundance of muons at sea level confirms this extended lifetime and therefore experimentally verifies time dilation.

Marking notes. 1 mark for stating the classical prediction (muons should decay before reaching sea level). 1 mark for correct calculation of dilated lifetime (Δt = 7 × 2.2 = 15.4 µs). 1 mark for calculating the increased travel distance consistent with reaching sea level. 1 mark for linking the observed sea-level detection to confirmation of time dilation (quantitative agreement).

Sample response. In 1972, Hafele and Keating flew four caesium atomic clocks on commercial aircraft eastward and westward around the world and compared them to reference clocks at the US Naval Observatory. The flying clocks registered small but measurable time differences relative to the ground clocks. Eastward-flying clocks lost time (ran slow) and westward-flying clocks gained time (ran fast) relative to ground clocks, consistent in sign and approximate magnitude with the combined predictions of special relativistic time dilation (speed effect) and general relativistic gravitational time dilation (altitude effect). This demonstrated that time dilation is a real, measurable physical effect even at aircraft speeds.

Marking notes. 1 mark for correctly describing the setup (atomic clocks on commercial aircraft, compared to ground clocks). 1 mark for stating the direction of the effect (eastward lost time, westward gained time, or general: flying clocks differ from ground clocks). 1 mark for linking the result to special relativistic time dilation (or noting both SR and GR contributions) as experimental confirmation.

Sample response. The student is incorrect. While it is true that, during the outbound and inbound legs, each twin sees the other’s clock run slow (reciprocal time dilation in special relativity), the situation is NOT symmetric overall. The travelling twin must decelerate, turn around, and re-accelerate at the distant star, changing inertial frames. The Earth-bound twin remains in a single inertial frame throughout the journey. This acceleration breaks the symmetry. General relativity, which handles accelerating frames, confirms that the accelerated twin (the traveller) accumulates less proper time and is objectively younger upon reunion. The Earth twin is never accelerated, so there is an unambiguous asymmetry.

Marking notes. 1 mark for identifying the flaw (the student incorrectly assumes symmetry applies throughout the full journey). 1 mark for identifying the key asymmetry (the travelling twin accelerates / changes inertial frames at the turnaround; Earth twin does not). 1 mark for stating the consequence (travelling twin accumulates less proper time / is objectively younger).

Sample response. Special relativistic effect: GPS satellites orbit at ~14,000 km/h. Their speed relative to Earth makes their clocks tick slower than ground clocks by approximately 7 µs per day (time dilation due to velocity). General relativistic effect: satellites are ~20,200 km above Earth’s surface, where the gravitational field is weaker, causing their clocks to tick faster than ground clocks by approximately 45 µs per day (gravitational time dilation). The net effect is that satellite clocks run fast by ~38 µs per day relative to ground clocks. Without correction, GPS positions would drift by approximately 10 km per day.

Marking notes. 1 mark for correctly describing the SR effect (speed makes clocks run slow, ~7 µs/day). 1 mark for correctly describing the GR effect (altitude / weaker gravity makes clocks run fast, ~45 µs/day). 1 mark for stating the net correction (≈ +38 µs/day faster for satellites, or equivalent) and/or the consequence if uncorrected (~10 km/day drift).

Sample response (a): γ. γ = 1/√(1 − v²/c²) = 1/√(1 − 0.64) = 1/√0.36 = 1/0.6 = 5/3 ≈ 1.667.

Sample response (b): Earth time. Round-trip distance = 2 × 8.0 = 16.0 light-years. At 0.80c: Δt = 16.0/0.80 = 20.0 years (Earth frame). Marking: 1 mark for correct round-trip distance; 1 mark for correct Δt = 20.0 years.

Sample response (c): proper time. Δt⊂0; = Δt / γ = 20.0 / (5/3) = 20.0 × 0.6 = 12.0 years.

Sample response (d): age difference and asymmetry. The Earth twin is 8.0 years older (20.0 − 12.0 = 8.0 years) [1]. The spacetime diagram shows the asymmetry: the Earth twin’s worldline is straight (vertical line at x = 0), traversing the maximum proper time. The astronaut’s worldline is kinked — it goes out diagonally at 0.8c then returns at 0.8c, forming a V-shape [1]. The kink represents the turnaround where the astronaut changes inertial frames by accelerating. A straight worldline always corresponds to the longest proper time (the “twin” paradox reflects that the longest straight worldline is the one that stays put) [1].

Marking notes. Part (a): 1 mark for γ = 5/3 with correct working. Part (b): 1 mark for correct round-trip distance; 1 mark for Δt = 20.0 yr. Part (c): 1 mark for Δt⊂0; = 12.0 yr. Part (d): 1 mark for age difference = 8.0 years; 1 mark for identifying the V-shape / kinked worldline; 1 mark for correctly linking the kink to the change of inertial frame / acceleration.

Sample response (a). Δt = γτ⊂0; = 20 × 26 = 520 ns.

Sample response (b). d = v × Δt ≈ c × Δt = 3.0 × 10⁸ × 520 × 10⁻⁹ = 156 m.

Sample response (c). The classical distance using proper lifetime would be c × τ⊂0; = 3.0 × 10⁸ × 26 × 10⁻⁹ ≈ 7.8 m. The pion travels 156 m rather than 7.8 m because time dilation extends its lifetime in the lab frame by a factor of γ = 20. From the pion’s own rest frame, this corresponds to length contraction of the lab (the pion sees the detector approaching at 0.999c over a contracted distance), but in the lab frame, the explanation is that the pion’s clock runs slow by γ, giving it more time to travel before decaying.

Marking notes. 1 mark for Δt = 520 ns. 1 mark for d = 156 m (accept 150–160 m). 1 mark for correctly attributing the greater distance to time dilation extending the lab-frame lifetime by γ = 20.

Sample response. The time dilation formula Δt = γΔt⊂0; makes quantitative predictions about the lifetimes of unstable particles and the rates of precision clocks, both of which have been tested experimentally.

The cosmic ray muon observations are a compelling piece of evidence. Muons produced at ~15 km altitude by cosmic-ray interactions have a proper mean lifetime of 2.2 µs. At ~0.99c (γ ≈ 7), classical physics predicts they travel ~650 m before decaying, far short of sea level. Yet muons are routinely detected at sea level in large numbers. Applying Δt = γτ⊂0; = 7 × 2.2 = 15.4 µs gives a lab-frame range of ~4.6 km; at higher velocities (γ ≈ 22), the formula gives ~14 km, consistent with the full atmospheric depth. The strength of this evidence lies in its quantitative agreement with the formula across different muon velocities measured at different altitudes, ruling out coincidence. A limitation is that determining the exact muon velocity requires separate measurement, and the comparison relies on particle physics models for muon production altitude and flux; systematic uncertainties can be significant. It is difficult to isolate a test of the time dilation formula alone without assuming other aspects of special relativity (such as the Lorentz transformation of decay rates).

The Hafele-Keating experiment provides direct evidence using macroscopic clocks. Four caesium atomic clocks were flown on commercial aircraft around the world and compared to ground clocks. The observed time differences (−59 ± 10 ns for eastward flights; +273 ± 7 ns for westward) were broadly consistent in sign with relativistic predictions, though the eastward result showed a large discrepancy (~47%) between the predicted −40 ns and observed −59 ns. A key strength is that this directly measures clock-rate differences at non-relativistic speeds (~900 km/h), confirming that time dilation is not limited to near-light-speed motion. A significant limitation is that the experiment cannot cleanly separate special relativistic effects from general relativistic (gravitational time dilation) effects: the net observed shift is the sum of both, and isolating the SR contribution requires a theoretical model. The experimental uncertainties (aircraft vibration, temperature, non-inertial flight paths) also reduce precision. The eastward discrepancy suggests either experimental error or that the SR/GR separation was imperfect.

Together, the two experiments complement each other. The muon data test the formula at extreme relativistic speeds (γ >> 1) and show the formula is accurate by orders of magnitude in predicting particle ranges. The Hafele-Keating experiment confirms the effect at low speeds with real clocks, but with lower precision and complicated by GR effects. Neither experiment alone provides a clean, unambiguous test of the SR time dilation formula in isolation; however, their agreement in sign, the correct qualitative behaviour, and quantitative consistency within uncertainties (especially for muons) constitute convincing evidence. Subsequent experiments (e.g. time dilation of stored muons in CERN rings at γ up to 29, agreeing with predictions to 0.2%) provide far more precise confirmation of Δt = γΔt⊂0;.

Marking criteria (7 marks). 1 = correctly states and applies the muon classical prediction (muons should not reach sea level without time dilation, with reference to ~650 m range). 1 = correctly evaluates a strength of muon evidence (quantitative agreement with Δt = γτ⊂0; across velocities). 1 = correctly evaluates a limitation of muon evidence (requires model for muon production / cannot isolate formula alone). 1 = correctly describes and evaluates a strength of Hafele-Keating (direct measurement of clock rate differences / confirms effect at low speed). 1 = correctly evaluates a limitation of Hafele-Keating (cannot cleanly separate SR from GR / experimental uncertainties). 1 = synthesises both experiments, explaining how they are complementary (different speed regimes, different types of clocks). 1 = makes an explicit evaluative judgement about the degree to which each independently (or jointly) confirms Δt = γΔt⊂0;, with reference to subsequent experiments or precision.