Physics • Year 12 • Module 7 • Lesson 12
Time Dilation
Lock in core vocabulary, the light clock derivation, and the meaning of proper time before tackling harder calculation problems.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from this list: proper time, dilated time, Lorentz factor, light clock, time dilation, inertial frame, special relativity, twin paradox, cosmic ray muon, Hafele-Keating experiment. 10 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The time interval between two events measured in the frame where both events occur at the same position — always the shortest possible measured time. | |
| 1.2 | The longer time interval measured by an observer moving relative to the clock; given by Δt = γΔt⊂0;. | |
| 1.3 | The quantity γ = 1 / √(1 − v²/c²), always ≥ 1, that relates measurements in different inertial frames. | |
| 1.4 | A thought-experiment device consisting of two mirrors with a bouncing photon; used to derive the time dilation formula from first principles. | |
| 1.5 | The phenomenon where a moving clock runs slower than an identical clock at rest, as measured by an observer in the rest frame. | |
| 1.6 | A reference frame in which Newton’s first law holds — an observer not undergoing acceleration. | |
| 1.7 | Einstein’s theory based on two postulates: the laws of physics are the same in all inertial frames, and the speed of light in a vacuum is constant for all observers. | |
| 1.8 | A thought experiment in which one twin travels at near-light-speed and returns younger than the Earth-bound twin; resolved by noting the traveller changes inertial frames. | |
| 1.9 | An unstable particle produced when cosmic rays strike the upper atmosphere; its detection at sea level is direct experimental evidence for time dilation. | |
| 1.10 | A 1972 experiment in which atomic clocks flown on commercial aircraft showed time differences consistent with relativistic predictions. |
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 Proper time is always measured by the observer on Earth. T / F
2.2 Time dilation only becomes significant at speeds very close to the speed of light. T / F
2.3 In the light clock thought experiment, a moving observer sees the photon travel a longer diagonal path, and since the speed of light is constant, the time for one tick is longer. T / F
2.4 Dilated time is always less than proper time. T / F
2.5 Cosmic ray muons reaching Earth’s surface provide evidence for time dilation because in Earth’s frame their lifetime is extended by the Lorentz factor. T / F
2.6 In the twin paradox, the situation is symmetric: each twin has equal claim to being younger when they reunite. 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:
acceleration · constant · dilated · gamma · longer · proper · shorter · slower
In special relativity, the speed of light in a vacuum is ___________ for all inertial observers. A clock moving relative to an observer ticks ___________ than an identical clock at rest. The time measured in the clock’s own rest frame is called the ___________ time, Δt⊂0;. The time measured in another inertial frame is called the ___________ time, Δt. The dilated time is always ___________ than the proper time. The ratio Δt / Δt⊂0; equals the Lorentz factor, ___________. The twin paradox is resolved because the travelling twin undergoes ___________ when turning around, breaking the symmetry. In Earth’s frame, a cosmic ray muon’s lifetime appears ___________ than its proper lifetime, allowing it to reach the surface.
4. Function recall
Answer each question in 1–2 sentences using precise terms from the lesson. 8 marks (2 each)
4.1 What is the defining criterion for labelling a time measurement as “proper time”?
4.2 How does the light clock thought experiment use the constancy of the speed of light to derive time dilation?
4.3 Why do GPS satellite clocks require relativistic corrections, and in which direction does special relativity shift the clock rate?
4.4 What experimental observation in the Hafele-Keating experiment confirmed special relativistic time dilation?
5. 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. “is measured by”, “is always less than”, “uses”). Aim for at least 6 labelled arrows. 6 marks (1 per valid labelled arrow)
Supplied terms: proper time · Lorentz factor · dilated time · light clock · velocity · inertial frame.
Q1 — Term–definition match
1.1 proper time • 1.2 dilated time • 1.3 Lorentz factor • 1.4 light clock • 1.5 time dilation • 1.6 inertial frame • 1.7 special relativity • 1.8 twin paradox • 1.9 cosmic ray muon • 1.10 Hafele-Keating experiment.
Q2 — True / false with correction
2.1 False. Proper time is measured by a clock present at both events — i.e. the clock in the frame where the two events occur at the same position. This is not necessarily the Earth observer; it is whichever clock “travels with” the events.
2.2 False. Time dilation occurs at ANY relative velocity; the Lorentz factor is greater than 1 for all v > 0. At everyday speeds the effect is unmeasurably small but real. The Hafele-Keating experiment detected it at aircraft speeds (~900 km/h).
2.3 True.
2.4 False. Dilated time is always greater than proper time (Δt = γΔt⊂0; with γ ≥ 1, so Δt ≥ Δt⊂0;). Proper time is the shortest possible measured interval.
2.5 True.
2.6 False. The situation is NOT symmetric. The travelling twin must accelerate to turn around, changing inertial frames. The Earth-bound twin remains in one inertial frame throughout. Only the travelling twin is younger upon reunion.
Q3 — Cloze paragraph
In order: constant / slower / proper / dilated / longer / gamma / acceleration / longer.
Q4.1 — Defining criterion for proper time
Proper time (Δt⊂0;) is the time interval between two events measured by a single clock that is present at both events — i.e. the clock is at rest in the frame where the two events occur at the same position. It is the shortest time measured by any observer.
Q4.2 — Light clock derivation
In the rest frame, a photon bounces straight up and down: Δt⊂0; = 2d/c. When the clock moves sideways at speed v, the photon must travel a longer diagonal path to keep up with the moving mirror. Since the speed of light c is the same for all observers, the longer path means more time per tick: Δt = Δt⊂0; / √(1 − v²/c²) = γΔt⊂0;.
Q4.3 — GPS relativistic corrections
GPS satellites orbit at ~14,000 km/h; their speed causes special relativistic time dilation that makes satellite clocks tick ~7 µs/day slower than ground clocks. Without correction, GPS position would drift ~10 km/day. Special relativity alone slows the satellite clocks (correction makes them run faster in software).
Q4.4 — Hafele-Keating result
Atomic clocks flown on commercial aircraft around the world returned with measurably different readings from identical clocks left on the ground. The aircraft clocks lost time relative to ground clocks (special relativistic effect), in quantitative agreement with relativistic predictions within experimental uncertainty.
Q5 — Sample concept map
Correct maps should include arrows such as:
- velocity — determines → Lorentz factor
- Lorentz factor — multiplies → proper time → dilated time
- light clock — illustrates derivation of → proper time
- dilated time — is measured in a different → inertial frame
- inertial frame — sets the reference for → velocity
- proper time — is always less than → dilated time
Award 1 mark per valid labelled arrow (minimum 6, maximum 6 marked).