IB · PHYSICS HL

Physics: Higher Level

Special relativity — HL Theme A

Name: ____________________Date: October 10, 2026

1.
State the two postulates of Einstein's special theory of relativity.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that the laws of physics are the same in all inertial (non-accelerating) reference frames. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or the observer. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The second postulate (invariance of the speed of light) is the one that leads directly to time dilation and length contraction — it is the genuinely surprising, non-classical claim.
Marking points
- States that the laws of physics are the same in all inertial (non-accelerating) reference frames.
- States that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or the observer.
Examiner tip: The second postulate (invariance of the speed of light) is the one that leads directly to time dilation and length contraction — it is the genuinely surprising, non-classical claim.
2.
Marking analysis: A learner attempts the following task: “State the two postulates of Einstein's special theory of relativity.” Their response addresses only this point: “States that the laws of physics are the same in all inertial (non-accelerating) reference frames.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that the laws of physics are the same in all inertial (non-accelerating) reference frames. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or the observer. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that the laws of physics are the same in all inertial (non-accelerating) reference frames.
- Identifies the missing requirement: States that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or the observer.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
3.
Define proper time, and state which observer measures the proper time interval between two events.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: Defines proper time as the time interval between two events measured by an observer for whom both events occur at the same location. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that the proper time is measured by the observer moving with the events (e.g. a clock travelling with a moving spacecraft), and is the shortest time interval measured by any observer. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Proper time is always the shortest time measured between two events by any observer — all other observers, in relative motion, measure a longer (dilated) time.
Marking points
- Defines proper time as the time interval between two events measured by an observer for whom both events occur at the same location.
- States that the proper time is measured by the observer moving with the events (e.g. a clock travelling with a moving spacecraft), and is the shortest time interval measured by any observer.
Examiner tip: Proper time is always the shortest time measured between two events by any observer — all other observers, in relative motion, measure a longer (dilated) time.
4.
Marking analysis: A learner attempts the following task: “Define proper time, and state which observer measures the proper time interval between two events.” Their response addresses only this point: “Defines proper time as the time interval between two events measured by an observer for whom both events occur at the same location.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Defines proper time as the time interval between two events measured by an observer for whom both events occur at the same location. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that the proper time is measured by the observer moving with the events (e.g. a clock travelling with a moving spacecraft), and is the shortest time interval measured by any observer. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Defines proper time as the time interval between two events measured by an observer for whom both events occur at the same location.
- Identifies the missing requirement: States that the proper time is measured by the observer moving with the events (e.g. a clock travelling with a moving spacecraft), and is the shortest time interval measured by any observer.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
5.
A spacecraft clock measures a proper time interval of 10.0 s. The spacecraft moves at 0.80c relative to an observer on Earth. Calculate the time interval measured by the Earth observer, using the Lorentz factor γ = 1/√(1 − v²/c²).
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates the Lorentz factor: γ = 1/√(1 − 0.80²) = 1/√(0.36) = 1/0.6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Develop this part of the answer: Obtains γ ≈ 1.67. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Work through this mathematical step: Uses time dilation Δt = γΔt₀. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes 1.67 × 10.0 to obtain Δt ≈ 16.7 s. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The dilated time (measured by the observer for whom the clock is moving) is always longer than the proper time — the Lorentz factor γ is always ≥ 1.
Marking points
- Calculates the Lorentz factor: γ = 1/√(1 − 0.80²) = 1/√(0.36) = 1/0.6.
- Obtains γ ≈ 1.67.
- Uses time dilation Δt = γΔt₀.
- Substitutes 1.67 × 10.0 to obtain Δt ≈ 16.7 s.
Examiner tip: The dilated time (measured by the observer for whom the clock is moving) is always longer than the proper time — the Lorentz factor γ is always ≥ 1.
6.
Marking analysis: A learner attempts the following task: “A spacecraft clock measures a proper time interval of 10.0 s. The spacecraft moves at 0.80c relative to an observer on Earth. Calculate the time interval measured by the Earth observer, using the Lorentz factor γ = 1/√(1 − v²/c²).” Their response addresses only this point: “Calculates the Lorentz factor: γ = 1/√(1 − 0.80²) = 1/√(0.36) = 1/0.6.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Calculates the Lorentz factor: γ = 1/√(1 − 0.80²) = 1/√(0.36) = 1/0.6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Obtains γ ≈ 1.67. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Uses time dilation Δt = γΔt₀. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Substitutes 1.67 × 10.0 to obtain Δt ≈ 16.7 s. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Calculates the Lorentz factor: γ = 1/√(1 − 0.80²) = 1/√(0.36) = 1/0.6.
- Identifies the missing requirement: Obtains γ ≈ 1.67.
- Identifies the missing requirement: Uses time dilation Δt = γΔt₀.
- Identifies the missing requirement: Substitutes 1.67 × 10.0 to obtain Δt ≈ 16.7 s.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
7.
Define proper length, and state what happens to the measured length of a moving object, as observed from a frame in which the object is moving.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: Defines proper length as the length of an object measured in the reference frame in which the object is at rest. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that the length measured in a frame where the object is moving (along the direction of motion) is contracted (shorter) compared to the proper length. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Length contraction only occurs along the direction of relative motion — dimensions perpendicular to the motion are unaffected.
Marking points
- Defines proper length as the length of an object measured in the reference frame in which the object is at rest.
- States that the length measured in a frame where the object is moving (along the direction of motion) is contracted (shorter) compared to the proper length.
Examiner tip: Length contraction only occurs along the direction of relative motion — dimensions perpendicular to the motion are unaffected.
8.
Marking analysis: A learner attempts the following task: “Define proper length, and state what happens to the measured length of a moving object, as observed from a frame in which the object is moving.” Their response addresses only this point: “Defines proper length as the length of an object measured in the reference frame in which the object is at rest.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Defines proper length as the length of an object measured in the reference frame in which the object is at rest. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that the length measured in a frame where the object is moving (along the direction of motion) is contracted (shorter) compared to the proper length. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Defines proper length as the length of an object measured in the reference frame in which the object is at rest.
- Identifies the missing requirement: States that the length measured in a frame where the object is moving (along the direction of motion) is contracted (shorter) compared to the proper length.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
9.
A spacecraft has a proper length of 120 m and travels at 0.60c relative to an observer on Earth. Calculate the length of the spacecraft as measured by the Earth observer.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates the Lorentz factor: γ = 1/√(1 − 0.60²) = 1/√(0.64) = 1/0.8 = 1.25. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Uses length contraction L = L₀/γ. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes 120/1.25. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains L = 96 m. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Length contraction (L = L₀/γ) and time dilation (Δt = γΔt₀) use the Lorentz factor in inverse ways — one divides by γ, the other multiplies — a common point of confusion.
Marking points
- Calculates the Lorentz factor: γ = 1/√(1 − 0.60²) = 1/√(0.64) = 1/0.8 = 1.25.
- Uses length contraction L = L₀/γ.
- Substitutes 120/1.25.
- Obtains L = 96 m.
Examiner tip: Length contraction (L = L₀/γ) and time dilation (Δt = γΔt₀) use the Lorentz factor in inverse ways — one divides by γ, the other multiplies — a common point of confusion.
10.
Marking analysis: A learner attempts the following task: “A spacecraft has a proper length of 120 m and travels at 0.60c relative to an observer on Earth. Calculate the length of the spacecraft as measured by the Earth observer.” Their response addresses only this point: “Calculates the Lorentz factor: γ = 1/√(1 − 0.60²) = 1/√(0.64) = 1/0.8 = 1.25.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Calculates the Lorentz factor: γ = 1/√(1 − 0.60²) = 1/√(0.64) = 1/0.8 = 1.25. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Uses length contraction L = L₀/γ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes 120/1.25. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains L = 96 m. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Calculates the Lorentz factor: γ = 1/√(1 − 0.60²) = 1/√(0.64) = 1/0.8 = 1.25.
- Identifies the missing requirement: Uses length contraction L = L₀/γ.
- Identifies the missing requirement: Substitutes 120/1.25.
- Identifies the missing requirement: Obtains L = 96 m.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
11.
Explain why no object with mass can be accelerated to reach or exceed the speed of light, using the relativistic mass-energy relationship.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that as an object's speed approaches c, the energy required to accelerate it further increases without bound. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Work through this mathematical step: States that this is because the relativistic kinetic energy (derived from E = γmc² − mc²) diverges to infinity as v approaches c, since γ approaches infinity. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Develop this part of the answer: States that since infinite energy would be required to reach c, no massive object can ever be accelerated to reach or exceed the speed of light. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: This is why the speed of light is described as a universal speed limit for anything with mass — only massless particles (like photons) can travel at exactly c.
Marking points
- States that as an object's speed approaches c, the energy required to accelerate it further increases without bound.
- States that this is because the relativistic kinetic energy (derived from E = γmc² − mc²) diverges to infinity as v approaches c, since γ approaches infinity.
- States that since infinite energy would be required to reach c, no massive object can ever be accelerated to reach or exceed the speed of light.
Examiner tip: This is why the speed of light is described as a universal speed limit for anything with mass — only massless particles (like photons) can travel at exactly c.
12.
Marking analysis: A learner attempts the following task: “Explain why no object with mass can be accelerated to reach or exceed the speed of light, using the relativistic mass-energy relationship.” Their response addresses only this point: “States that as an object's speed approaches c, the energy required to accelerate it further increases without bound.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that as an object's speed approaches c, the energy required to accelerate it further increases without bound. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that this is because the relativistic kinetic energy (derived from E = γmc² − mc²) diverges to infinity as v approaches c, since γ approaches infinity. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States that since infinite energy would be required to reach c, no massive object can ever be accelerated to reach or exceed the speed of light. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that as an object's speed approaches c, the energy required to accelerate it further increases without bound.
- Identifies the missing requirement: States that this is because the relativistic kinetic energy (derived from E = γmc² − mc²) diverges to infinity as v approaches c, since γ approaches infinity.
- Identifies the missing requirement: States that since infinite energy would be required to reach c, no massive object can ever be accelerated to reach or exceed the speed of light.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
13.
State Einstein's mass-energy equivalence relation, and calculate the energy equivalent of 1.0 g of mass. Use c = 3.00 × 10⁸ m s⁻¹.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States E = mc². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Converts mass to kilograms: 1.0 g = 1.0 × 10⁻³ kg. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes (1.0 × 10⁻³)(3.00 × 10⁸)² to obtain E = 9.0 × 10¹³ J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The enormous size of c² means even a tiny mass corresponds to a huge amount of energy — this is why nuclear reactions, which convert a small fraction of mass to energy, release so much more energy than chemical reactions.
Marking points
- States E = mc².
- Converts mass to kilograms: 1.0 g = 1.0 × 10⁻³ kg.
- Substitutes (1.0 × 10⁻³)(3.00 × 10⁸)² to obtain E = 9.0 × 10¹³ J.
Examiner tip: The enormous size of c² means even a tiny mass corresponds to a huge amount of energy — this is why nuclear reactions, which convert a small fraction of mass to energy, release so much more energy than chemical reactions.
14.
Marking analysis: A learner attempts the following task: “State Einstein's mass-energy equivalence relation, and calculate the energy equivalent of 1.0 g of mass. Use c = 3.00 × 10⁸ m s⁻¹.” Their response addresses only this point: “States E = mc².” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States E = mc². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Converts mass to kilograms: 1.0 g = 1.0 × 10⁻³ kg. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes (1.0 × 10⁻³)(3.00 × 10⁸)² to obtain E = 9.0 × 10¹³ J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States E = mc².
- Identifies the missing requirement: Converts mass to kilograms: 1.0 g = 1.0 × 10⁻³ kg.
- Identifies the missing requirement: Substitutes (1.0 × 10⁻³)(3.00 × 10⁸)² to obtain E = 9.0 × 10¹³ J.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
15.
Two events occur simultaneously in the reference frame of observer A, but at different locations. Explain, using relativity of simultaneity, why observer B, moving relative to A, may not measure the two events as simultaneous.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that simultaneity (whether two events occur at the same time) is not absolute in special relativity — it depends on the observer's reference frame. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that because the speed of light is the same for all observers, and the two events are at different locations, the light signals from each event take different times to reach a moving observer in a way that differs from a stationary one. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Develop this part of the answer: States that this results in observer B, in relative motion, measuring the two events as occurring at different times, even though observer A measures them as simultaneous. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Relativity of simultaneity is a direct consequence of the invariance of the speed of light combined with events occurring at different locations — it only matters when events are spatially separated.
Marking points
- States that simultaneity (whether two events occur at the same time) is not absolute in special relativity — it depends on the observer's reference frame.
- States that because the speed of light is the same for all observers, and the two events are at different locations, the light signals from each event take different times to reach a moving observer in a way that differs from a stationary one.
- States that this results in observer B, in relative motion, measuring the two events as occurring at different times, even though observer A measures them as simultaneous.
Examiner tip: Relativity of simultaneity is a direct consequence of the invariance of the speed of light combined with events occurring at different locations — it only matters when events are spatially separated.
16.
Marking analysis: A learner attempts the following task: “Two events occur simultaneously in the reference frame of observer A, but at different locations. Explain, using relativity of simultaneity, why observer B, moving relative to A, may not measure the two events as simultaneous.” Their response addresses only this point: “States that simultaneity (whether two events occur at the same time) is not absolute in special relativity — it depends on the observer's reference frame.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that simultaneity (whether two events occur at the same time) is not absolute in special relativity — it depends on the observer's reference frame. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that because the speed of light is the same for all observers, and the two events are at different locations, the light signals from each event take different times to reach a moving observer in a way that differs from a stationary one. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States that this results in observer B, in relative motion, measuring the two events as occurring at different times, even though observer A measures them as simultaneous. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that simultaneity (whether two events occur at the same time) is not absolute in special relativity — it depends on the observer's reference frame.
- Identifies the missing requirement: States that because the speed of light is the same for all observers, and the two events are at different locations, the light signals from each event take different times to reach a moving observer in a way that differs from a stationary one.
- Identifies the missing requirement: States that this results in observer B, in relative motion, measuring the two events as occurring at different times, even though observer A measures them as simultaneous.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
17.
A muon is created in the upper atmosphere and has a proper lifetime of 2.2 μs. Explain, using time dilation, how muons travelling close to the speed of light can be detected at the Earth's surface despite this short lifetime.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that from the Earth observer's frame, the muon's lifetime is time-dilated (appears longer) because the muon is moving at a speed close to c. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that this dilated lifetime is long enough for the muon to travel the distance through the atmosphere to reach the Earth's surface before decaying, even though its proper lifetime alone would not allow this. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Muon decay is one of the classic real experimental confirmations of special relativity's time dilation, since far more muons reach the ground than classical (non-relativistic) calculations would predict.
Marking points
- States that from the Earth observer's frame, the muon's lifetime is time-dilated (appears longer) because the muon is moving at a speed close to c.
- States that this dilated lifetime is long enough for the muon to travel the distance through the atmosphere to reach the Earth's surface before decaying, even though its proper lifetime alone would not allow this.
Examiner tip: Muon decay is one of the classic real experimental confirmations of special relativity's time dilation, since far more muons reach the ground than classical (non-relativistic) calculations would predict.
18.
Marking analysis: A learner attempts the following task: “A muon is created in the upper atmosphere and has a proper lifetime of 2.2 μs. Explain, using time dilation, how muons travelling close to the speed of light can be detected at the Earth's surface despite this short lifetime.” Their response addresses only this point: “States that from the Earth observer's frame, the muon's lifetime is time-dilated (appears longer) because the muon is moving at a speed close to c.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that from the Earth observer's frame, the muon's lifetime is time-dilated (appears longer) because the muon is moving at a speed close to c. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that this dilated lifetime is long enough for the muon to travel the distance through the atmosphere to reach the Earth's surface before decaying, even though its proper lifetime alone would not allow this. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that from the Earth observer's frame, the muon's lifetime is time-dilated (appears longer) because the muon is moving at a speed close to c.
- Identifies the missing requirement: States that this dilated lifetime is long enough for the muon to travel the distance through the atmosphere to reach the Earth's surface before decaying, even though its proper lifetime alone would not allow this.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
19.
Outline why, at everyday (non-relativistic) speeds, the effects of time dilation and length contraction are completely unnoticeable.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that for everyday speeds, v is extremely small compared to c, so v²/c² is extremely close to zero. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that this makes the Lorentz factor γ extremely close to 1, so the predicted time dilation and length contraction effects are negligibly small and unmeasurable in ordinary situations. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Relativistic effects only become significant when an object's speed is a substantial fraction of the speed of light (typically above about 0.1c) — this is why classical (Newtonian) mechanics works perfectly well for everyday life.
Marking points
- States that for everyday speeds, v is extremely small compared to c, so v²/c² is extremely close to zero.
- States that this makes the Lorentz factor γ extremely close to 1, so the predicted time dilation and length contraction effects are negligibly small and unmeasurable in ordinary situations.
Examiner tip: Relativistic effects only become significant when an object's speed is a substantial fraction of the speed of light (typically above about 0.1c) — this is why classical (Newtonian) mechanics works perfectly well for everyday life.
20.
Marking analysis: A learner attempts the following task: “Outline why, at everyday (non-relativistic) speeds, the effects of time dilation and length contraction are completely unnoticeable.” Their response addresses only this point: “States that for everyday speeds, v is extremely small compared to c, so v²/c² is extremely close to zero.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that for everyday speeds, v is extremely small compared to c, so v²/c² is extremely close to zero. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that this makes the Lorentz factor γ extremely close to 1, so the predicted time dilation and length contraction effects are negligibly small and unmeasurable in ordinary situations. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that for everyday speeds, v is extremely small compared to c, so v²/c² is extremely close to zero.
- Identifies the missing requirement: States that this makes the Lorentz factor γ extremely close to 1, so the predicted time dilation and length contraction effects are negligibly small and unmeasurable in ordinary situations.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
21.
A particle of mass 2.0 × 10⁻²⁷ kg moves at a speed of 0.90c. Calculate its relativistic momentum, given that the Lorentz factor at this speed is γ = 2.29. Use c = 3.00 × 10⁸ m s⁻¹.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses relativistic momentum p = γmv. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates v = 0.90 × 3.00 × 10⁸ = 2.70 × 10⁸ m s⁻¹. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes p = 2.29 × (2.0 × 10⁻²⁷) × (2.70 × 10⁸). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Develop this part of the answer: Obtains p ≈ 1.24 × 10⁻¹⁸ kg m s⁻¹. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Relativistic momentum includes the Lorentz factor γ, unlike classical momentum p = mv — at high speeds this makes momentum grow faster than speed alone would suggest, diverging as v approaches c.
Marking points
- Uses relativistic momentum p = γmv.
- Calculates v = 0.90 × 3.00 × 10⁸ = 2.70 × 10⁸ m s⁻¹.
- Substitutes p = 2.29 × (2.0 × 10⁻²⁷) × (2.70 × 10⁸).
- Obtains p ≈ 1.24 × 10⁻¹⁸ kg m s⁻¹.
Examiner tip: Relativistic momentum includes the Lorentz factor γ, unlike classical momentum p = mv — at high speeds this makes momentum grow faster than speed alone would suggest, diverging as v approaches c.
22.
Marking analysis: A learner attempts the following task: “A particle of mass 2.0 × 10⁻²⁷ kg moves at a speed of 0.90c. Calculate its relativistic momentum, given that the Lorentz factor at this speed is γ = 2.29. Use c = 3.00 × 10⁸ m s⁻¹.” Their response addresses only this point: “Uses relativistic momentum p = γmv.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Uses relativistic momentum p = γmv. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates v = 0.90 × 3.00 × 10⁸ = 2.70 × 10⁸ m s⁻¹. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes p = 2.29 × (2.0 × 10⁻²⁷) × (2.70 × 10⁸). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains p ≈ 1.24 × 10⁻¹⁸ kg m s⁻¹. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Uses relativistic momentum p = γmv.
- Identifies the missing requirement: Calculates v = 0.90 × 3.00 × 10⁸ = 2.70 × 10⁸ m s⁻¹.
- Identifies the missing requirement: Substitutes p = 2.29 × (2.0 × 10⁻²⁷) × (2.70 × 10⁸).
- Identifies the missing requirement: Obtains p ≈ 1.24 × 10⁻¹⁸ kg m s⁻¹.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
23.
A spacecraft moving at 0.50c relative to Earth fires a probe forward at a speed of 0.60c relative to the spacecraft. Using the relativistic velocity addition formula u = (v + u′)/(1 + vu′/c²), calculate the probe's speed relative to Earth, and explain why this is less than the value 1.10c predicted by simple (Galilean) addition.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Substitutes v = 0.50c and u′ = 0.60c into u = (v + u′)/(1 + vu′/c²), giving u = (0.50c + 0.60c)/(1 + 0.50 × 0.60). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the numerator = 1.10c and the denominator = 1.30. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains u = 1.10/1.30 c ≈ 0.846c. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Develop this part of the answer: Explains that relativistic velocity addition always yields a result less than c, consistent with no object exceeding the speed of light, unlike Galilean addition which incorrectly gives a value (1.10c) greater than c. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The relativistic velocity addition formula automatically prevents any combination of sub-light speeds from ever producing a result at or above c — this built-in ceiling is exactly what the classical formula lacks.
Marking points
- Substitutes v = 0.50c and u′ = 0.60c into u = (v + u′)/(1 + vu′/c²), giving u = (0.50c + 0.60c)/(1 + 0.50 × 0.60).
- Calculates the numerator = 1.10c and the denominator = 1.30.
- Obtains u = 1.10/1.30 c ≈ 0.846c.
- Explains that relativistic velocity addition always yields a result less than c, consistent with no object exceeding the speed of light, unlike Galilean addition which incorrectly gives a value (1.10c) greater than c.
Examiner tip: The relativistic velocity addition formula automatically prevents any combination of sub-light speeds from ever producing a result at or above c — this built-in ceiling is exactly what the classical formula lacks.
24.
Marking analysis: A learner attempts the following task: “A spacecraft moving at 0.50c relative to Earth fires a probe forward at a speed of 0.60c relative to the spacecraft. Using the relativistic velocity addition formula u = (v + u′)/(1 + vu′/c²), calculate the probe's speed relative to Earth, and explain why this is less than the value 1.10c predicted by simple (Galilean) addition.” Their response addresses only this point: “Substitutes v = 0.50c and u′ = 0.60c into u = (v + u′)/(1 + vu′/c²), giving u = (0.50c + 0.60c)/(1 + 0.50 × 0.60).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Substitutes v = 0.50c and u′ = 0.60c into u = (v + u′)/(1 + vu′/c²), giving u = (0.50c + 0.60c)/(1 + 0.50 × 0.60). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates the numerator = 1.10c and the denominator = 1.30. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains u = 1.10/1.30 c ≈ 0.846c. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Explains that relativistic velocity addition always yields a result less than c, consistent with no object exceeding the speed of light, unlike Galilean addition which incorrectly gives a value (1.10c) greater than c. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Substitutes v = 0.50c and u′ = 0.60c into u = (v + u′)/(1 + vu′/c²), giving u = (0.50c + 0.60c)/(1 + 0.50 × 0.60).
- Identifies the missing requirement: Calculates the numerator = 1.10c and the denominator = 1.30.
- Identifies the missing requirement: Obtains u = 1.10/1.30 c ≈ 0.846c.
- Identifies the missing requirement: Explains that relativistic velocity addition always yields a result less than c, consistent with no object exceeding the speed of light, unlike Galilean addition which incorrectly gives a value (1.10c) greater than c.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
25.
In the 'twin paradox', one twin travels on a high-speed round trip to a distant star while the other twin remains on Earth. Explain why, upon reunion, the travelling twin is found to have aged less than the twin who stayed on Earth, and state why this is not a true paradox despite each twin seeing the other's clock as running slow during the trip.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that, from Earth's frame, the travelling twin's clock runs slow (is time-dilated) throughout the journey due to their high speed. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that the situation is not symmetric because the travelling twin must accelerate or decelerate, e.g. to turn around at the distant star, changing reference frames, while the Earth twin remains in a single inertial frame throughout. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Develop this part of the answer: States that this acceleration breaks the symmetry between the twins, so special relativity, which strictly applies to inertial frames, predicts a genuine, calculable difference in elapsed proper time. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
5. Develop this part of the answer: States that the travelling twin is therefore found to have aged less upon reunion, a result that has been experimentally confirmed, e.g. using precise atomic clocks flown on aircraft. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The 'paradox' dissolves once you notice the asymmetry: only the travelling twin experiences acceleration (leaves an inertial frame), so the situation was never actually symmetric between the two twins in the first place.
Marking points
- States that, from Earth's frame, the travelling twin's clock runs slow (is time-dilated) throughout the journey due to their high speed.
- States that the situation is not symmetric because the travelling twin must accelerate or decelerate, e.g. to turn around at the distant star, changing reference frames, while the Earth twin remains in a single inertial frame throughout.
- States that this acceleration breaks the symmetry between the twins, so special relativity, which strictly applies to inertial frames, predicts a genuine, calculable difference in elapsed proper time.
- States that the travelling twin is therefore found to have aged less upon reunion, a result that has been experimentally confirmed, e.g. using precise atomic clocks flown on aircraft.
Examiner tip: The 'paradox' dissolves once you notice the asymmetry: only the travelling twin experiences acceleration (leaves an inertial frame), so the situation was never actually symmetric between the two twins in the first place.
26.
Marking analysis: A learner attempts the following task: “In the 'twin paradox', one twin travels on a high-speed round trip to a distant star while the other twin remains on Earth. Explain why, upon reunion, the travelling twin is found to have aged less than the twin who stayed on Earth, and state why this is not a true paradox despite each twin seeing the other's clock as running slow during the trip.” Their response addresses only this point: “States that, from Earth's frame, the travelling twin's clock runs slow (is time-dilated) throughout the journey due to their high speed.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that, from Earth's frame, the travelling twin's clock runs slow (is time-dilated) throughout the journey due to their high speed. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that the situation is not symmetric because the travelling twin must accelerate or decelerate, e.g. to turn around at the distant star, changing reference frames, while the Earth twin remains in a single inertial frame throughout. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States that this acceleration breaks the symmetry between the twins, so special relativity, which strictly applies to inertial frames, predicts a genuine, calculable difference in elapsed proper time. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: States that the travelling twin is therefore found to have aged less upon reunion, a result that has been experimentally confirmed, e.g. using precise atomic clocks flown on aircraft. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that, from Earth's frame, the travelling twin's clock runs slow (is time-dilated) throughout the journey due to their high speed.
- Identifies the missing requirement: States that the situation is not symmetric because the travelling twin must accelerate or decelerate, e.g. to turn around at the distant star, changing reference frames, while the Earth twin remains in a single inertial frame throughout.
- Identifies the missing requirement: States that this acceleration breaks the symmetry between the twins, so special relativity, which strictly applies to inertial frames, predicts a genuine, calculable difference in elapsed proper time.
- Identifies the missing requirement: States that the travelling twin is therefore found to have aged less upon reunion, a result that has been experimentally confirmed, e.g. using precise atomic clocks flown on aircraft.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
27.
A particle has rest mass 1.67 × 10⁻²⁷ kg and moves with a Lorentz factor γ = 1.50. Calculate its total relativistic energy and its relativistic kinetic energy. Use c = 3.00 × 10⁸ m s⁻¹.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates the rest energy: E₀ = mc² = 1.67 × 10⁻²⁷ × (3.00 × 10⁸)² ≈ 1.50 × 10⁻¹⁰ J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Uses total relativistic energy E = γmc². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains E = 1.50 × 1.50 × 10⁻¹⁰ ≈ 2.25 × 10⁻¹⁰ J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Uses relativistic kinetic energy KE = (γ − 1)mc². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Obtains KE = 0.50 × 1.50 × 10⁻¹⁰ ≈ 7.52 × 10⁻¹¹ J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Total relativistic energy always splits cleanly into rest energy (mc², present even at rest) plus kinetic energy ((γ−1)mc², present only due to motion) — checking that these two add back to the total is a good way to verify the calculation.
Marking points
- Calculates the rest energy: E₀ = mc² = 1.67 × 10⁻²⁷ × (3.00 × 10⁸)² ≈ 1.50 × 10⁻¹⁰ J.
- Uses total relativistic energy E = γmc².
- Obtains E = 1.50 × 1.50 × 10⁻¹⁰ ≈ 2.25 × 10⁻¹⁰ J.
- Uses relativistic kinetic energy KE = (γ − 1)mc².
- Obtains KE = 0.50 × 1.50 × 10⁻¹⁰ ≈ 7.52 × 10⁻¹¹ J.
Examiner tip: Total relativistic energy always splits cleanly into rest energy (mc², present even at rest) plus kinetic energy ((γ−1)mc², present only due to motion) — checking that these two add back to the total is a good way to verify the calculation.
28.
Marking analysis: A learner attempts the following task: “A particle has rest mass 1.67 × 10⁻²⁷ kg and moves with a Lorentz factor γ = 1.50. Calculate its total relativistic energy and its relativistic kinetic energy. Use c = 3.00 × 10⁸ m s⁻¹.” Their response addresses only this point: “Calculates the rest energy: E₀ = mc² = 1.67 × 10⁻²⁷ × (3.00 × 10⁸)² ≈ 1.50 × 10⁻¹⁰ J.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Calculates the rest energy: E₀ = mc² = 1.67 × 10⁻²⁷ × (3.00 × 10⁸)² ≈ 1.50 × 10⁻¹⁰ J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Uses total relativistic energy E = γmc². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains E = 1.50 × 1.50 × 10⁻¹⁰ ≈ 2.25 × 10⁻¹⁰ J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Uses relativistic kinetic energy KE = (γ − 1)mc². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Obtains KE = 0.50 × 1.50 × 10⁻¹⁰ ≈ 7.52 × 10⁻¹¹ J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Calculates the rest energy: E₀ = mc² = 1.67 × 10⁻²⁷ × (3.00 × 10⁸)² ≈ 1.50 × 10⁻¹⁰ J.
- Identifies the missing requirement: Uses total relativistic energy E = γmc².
- Identifies the missing requirement: Obtains E = 1.50 × 1.50 × 10⁻¹⁰ ≈ 2.25 × 10⁻¹⁰ J.
- Identifies the missing requirement: Uses relativistic kinetic energy KE = (γ − 1)mc².
- Identifies the missing requirement: Obtains KE = 0.50 × 1.50 × 10⁻¹⁰ ≈ 7.52 × 10⁻¹¹ J.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
29.
A distant galaxy is moving away from Earth at a significant fraction of the speed of light. State and explain, using the relativistic Doppler effect, what happens to the observed wavelength of light received from this galaxy.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that the observed wavelength is increased (redshifted) compared to the wavelength emitted by the source. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: Explains that because the source is moving away from the observer, successive wave crests are emitted from increasingly greater distances, so they take progressively longer to reach the observer, stretching the observed wavelength. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Develop this part of the answer: States that the size of this redshift increases with the galaxy's recession speed, and can be used to determine that speed. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Redshift of distant galaxies' light is one of the key pieces of observational evidence for the expansion of the universe — the greater the redshift, the faster a galaxy is generally receding.
Marking points
- States that the observed wavelength is increased (redshifted) compared to the wavelength emitted by the source.
- Explains that because the source is moving away from the observer, successive wave crests are emitted from increasingly greater distances, so they take progressively longer to reach the observer, stretching the observed wavelength.
- States that the size of this redshift increases with the galaxy's recession speed, and can be used to determine that speed.
Examiner tip: Redshift of distant galaxies' light is one of the key pieces of observational evidence for the expansion of the universe — the greater the redshift, the faster a galaxy is generally receding.
30.
Marking analysis: A learner attempts the following task: “A distant galaxy is moving away from Earth at a significant fraction of the speed of light. State and explain, using the relativistic Doppler effect, what happens to the observed wavelength of light received from this galaxy.” Their response addresses only this point: “States that the observed wavelength is increased (redshifted) compared to the wavelength emitted by the source.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that the observed wavelength is increased (redshifted) compared to the wavelength emitted by the source. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Explains that because the source is moving away from the observer, successive wave crests are emitted from increasingly greater distances, so they take progressively longer to reach the observer, stretching the observed wavelength. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States that the size of this redshift increases with the galaxy's recession speed, and can be used to determine that speed. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that the observed wavelength is increased (redshifted) compared to the wavelength emitted by the source.
- Identifies the missing requirement: Explains that because the source is moving away from the observer, successive wave crests are emitted from increasingly greater distances, so they take progressively longer to reach the observer, stretching the observed wavelength.
- Identifies the missing requirement: States that the size of this redshift increases with the galaxy's recession speed, and can be used to determine that speed.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.