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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Identify the given values: l 0 = 100 m, v = 0.8c, c = 3 × 10 8 m/s.
- S2
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.)
- S3
Use the length contraction formula: l = l 0 × √(1 - v 2 /c 2 ) = l 0 /γ = 100 m × 0.6 = 60 m.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Identify the given values: l 0 = 100 m, v = 0.8c, c = 3 × 10 8 m/s.
- S2
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.)
- S3
Use the length contraction formula: l = l 0 × √(1 - v 2 /c 2 ) = l 0 /γ = 100 m × 0.6 = 60 m.
Final answer: The observed length of the spaceship is 60 meters.
Common mistakes
- Forgetting to use the speed of light (c) in the formula — Always include the speed of light (c) when calculating the Lorentz factor and length contraction.
- Misinterpreting the proper length and observed length — 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 — Ensure that you always square the velocity (v) when calculating the Lorentz factor.
- Using the wrong units for speed — 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 — 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 — 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
- Full worked solution (all marking points)4 marks