A-Level · Physics · AQA · Mark scheme decoded
AQA A-Level Physics: Length Contraction in Special Relativity — mark scheme explained
The short answer
Special relativity, introduced by Albert Einstein, fundamentally changed our understanding of space and time. One of the key predictions of special relativity is length contraction, which describes how objects moving at high speeds relative to an observer appear shorter along the direction of motion.
The question
A spaceship is traveling at a speed of 0.8c relative to an observer on Earth. The proper length of the spaceship is 100 meters. Calculate the observed length of the spaceship.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For calculation questions, marks are typically awarded for identifying the correct formula, substituting the given values correctly, performing the necessary calculations, and providing the final answer with appropriate units. For explanation questions, marks are awarded for clarity, accuracy, and completeness of the response.
What the command words demand
- Calculate
- Perform a numerical calculation using the given data and formulas.
- Determine
- Find the value of a specific quantity or variable.
- Explain
- Provide a clear and concise explanation, including relevant concepts and principles.
- Describe
- Give a detailed account of a phenomenon or process.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate approximately 5-7 minutes to solve a typical length contraction problem in an exam setting.
- Identify the given values: l 0 = 100 m, v = 0.8c, c = 3 × 10 8 m/s.0 marks
- Calculate the contraction factor √(1 - v 2 /c 2 ) = √(1 - (0.8c) 2 /c 2 ) = √(1 - 0.64) = √0.36 = 0.6. (This equals 1/γ; the Lorentz factor itself is γ = 1/0.6 ≈ 1.67.)2 marks
- Use the length contraction formula: l = l 0 × √(1 - v 2 /c 2 ) = l 0 /γ = 100 m × 0.6 = 60 m.2 marks
Final answer: The observed length of the spaceship is 60 meters.
Work through every step correctly and you earn all 4 marks.
Another worked example
A muon has a proper lifetime of 2.2 microseconds when at rest. If it travels at a speed of 0.99c, calculate the distance it can travel before decaying as observed by an Earth-based observer.
- Identify the given values: t 0 = 2.2 μs, v = 0.99c, c = 3 × 10 8 m/s.0 marks
- Calculate the Lorentz factor (γ): γ = 1 / √(1 - v 2 /c 2 ) = 1 / √(1 - (0.99) 2 ) = 1 / √0.0199 ≈ 7.09.2 marks
- Calculate the observed (dilated) lifetime t: t = γ × t 0 = 7.09 × 2.2 μs ≈ 15.6 μs. (The observed lifetime must be LONGER than the proper lifetime.)2 marks
- Calculate the distance traveled d: d = v × t = 0.99c × 15.6 μs = 0.99 × 3 × 10 8 m/s × 15.6 × 10 -6 s ≈ 4630 meters.2 marks
Final answer: The distance the muon can travel before decaying is approximately 4630 meters (about 4.6 km).
Work through every step correctly and you earn all 6 marks.
Common mistakes
Forgetting to use the speed of light (c) in the formula
Why it happens: Students may overlook the importance of using the speed of light in the Lorentz factor, leading to incorrect calculations.
Fix: Always include the speed of light (c) when calculating the Lorentz factor and length contraction.
Misinterpreting the proper length and observed length
Why it happens: Students may confuse which length is the proper length and which is the observed length, leading to incorrect application of the formula.
Fix: Clearly identify l 0 as the proper length (at rest) and l as the observed length (in motion).
Failing to square the velocity (v 2 ) in the Lorentz factor
Why it happens: Students may forget to square the velocity, leading to incorrect values for the Lorentz factor and observed length.
Fix: Ensure that you always square the velocity (v) when calculating the Lorentz factor.
Using the wrong units for speed
Why it happens: Students may use incorrect units for the speed of light or the object's speed, leading to dimensional inconsistencies.
Fix: Always use consistent units, such as meters per second (m/s) for both the speed of light and the object's speed.
Forgetting to take the square root in the Lorentz factor
Why it happens: Students may forget to apply the square root, leading to incorrect values for the Lorentz factor and observed length.
Fix: Ensure that you always take the square root of (1 - v 2 /c 2 ) when calculating the Lorentz factor.
Confusing time dilation with length contraction
Why it happens: Students may mix up the concepts of time dilation and length contraction, leading to incorrect application of formulas.
Fix: Understand that time dilation affects the passage of time, while length contraction affects the measurement of length. Use the appropriate formula for each concept.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Length Contraction | Apply the length contraction formula to find the observed length of a moving object. | 4 |
| Relativistic Time Dilation | Use time dilation to find the distance a fast-moving muon travels before decaying. | 6 |
| Length Contraction Calculation | Use the length contraction formula to find the observed length of a moving rod. | 4 |
| Length Contraction Calculation | Calculate the contracted distance measured by astronauts moving at relativistic speed relative to Earth. | 4 |
| Total across these question types | 18 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.