AQA A-Level Physics Paper 3 (3BD), June 2025: Question 4

9 marks · Hard difficulty · Extended Answer

Use special relativity concepts, including time dilation, length contraction, and the invariance of the speed of light, to determine journey times, cargo fit, and signal speed.

Practise this question

Question

Question 04 contains three parts about special relativity. In 04.1, spaceship A travels at 0.50c from planet O to P. The distance measured on P is 1.4 x 10^14 m, and food cargo must experience a journey time of no more than 10 days; students must determine if it delivers in time (4 marks). In 04.2, cargo length at rest on P is 210 m, and the length of moving spaceship A is measured as 200 m by an observer on P; students must determine if the cargo will fit into the hold of A (4 marks). In 04.3, Figure 5 shows spaceship A moving right at 0.50c and spaceship B moving left at 0.30c towards each other; students select the speed of a radio signal sent from B to A from options: 0.20c, 0.40c, 0.80c, c, or 1.8c (1 mark).
Question text

04 In a science-fiction novel, spaceship A transports cargo from planet O to planet P.

A travels at a constant speed of 0.50c in the reference frame of an observer on P.

04.1 The observer on P measures the distance from O to P as 1.4 × 1014 m.

Some of the cargo is food and must experience a journey time of no more

than 10 days.

Determine whether A is able to deliver the cargo within this time.

[4 marks]

04.2 The observer on P measures the length of a cargo at rest on P to be 210 m.

There is a cargo hold along the full length of A.

The observer on P measures the length of A to be 200 m in the direction of travel

while the spaceship is moving at 0.50c.

Determine whether the cargo will fit into the cargo hold of A.

[4 marks]

04.3 Spaceships A and B are travelling towards each other along the same straight line.

The speed of B is 0.30c with respect to P.

Figure 5 shows the situation.

Figure 5

B sends a radio signal to A.

What is the speed of the radio signal measured by A?

Tick ( ) one box.

[1 mark]

0.20c

0.40c

0.80c

c

1.8c

Mark scheme

Show the mark scheme Mark scheme for Question 04. For 04.1 (4 marks): uses time dilation or length contraction formula with proper time/length, substitutes into v = s/t within a consistent reference frame, calculates proper time (around 9.3 or 9.4 days) or dilated time (around 10.8 days), and compares to 10 days to conclude it can deliver in time. For 04.2 (4 marks): calculates the proper length of the spaceship hold (231 m) or contracted cargo length (182 m) using the Lorentz factor (sqrt(1 - 0.5^2) = 0.866), and compares to conclude the cargo will fit. For 04.3 (1 mark): the speed of the radio signal is c, with guidance stating the speed of light is invariant.

Question Answers Additional comments/Guidance Mark AO

𝑡 or 𝑡 or 𝑙 or 𝑙 identified and substituted into 𝑡 = 𝑙 = 1.4 × 1014, 𝑡 = 10 (days) or 864000 (s),

04.1 0 0 0 0 4 AO3

𝑡 𝑣2 𝑠

0 or 𝑙 = 𝑙 √1 − ✓ 𝑙 or 𝑡 from the result of 𝑣 = for their 𝑣 (allow 𝑡 in days)

20 𝑐2 𝑡

√ 𝑣

1− 2

𝑐

Identification can be explicit or inferred from substitution.

Condone missing squares on 𝑣 and 𝑐 in MP1.

s

Substitution using v = with their substituted 𝑣: either 𝑠 = 1.4 × 1014 or 𝑡 = 8.64 × 105 𝑠 with their answer from a

t

values in the same reference frame✓ relativistic calculation (but not with each other).

1.4×1014

Do not allow 𝑣 = 10(×24×60×60) or use of v = c for MP2 and MP4.

𝑠 or 𝑡: 𝑣 = 0.5𝑐 and one of 𝑠 = 1.4 × 1014, 𝑡 = 8.64 × 105 𝑠 or their

answer from a relativistic calculation

All correct shaded value(s) from any row seen ✓

– A-LEVEL PHYSICS – –

0.52 c2

04.2 (1.5×108)2 Condone MP1 if 𝐿 and 𝐿 are mixed up. 4 AO3

Use of 1− 0.52 or 1− or√1 − or √1 − 0.25 0

2 (3×108)2

c Squares must be seen in substitution for MP1

√3 but allow MP2 if missing.

0.75 or 0.866 or with lengths ✓

v

Correct substitution into l = l0 1− 2 gains first

c

2 marks

l = 200 (m) or l0 = 210 (m) identified ✓ 2 2

e.g. 𝑙 = 210√1 − 0.5 or 200 = 𝑙0√1 − 0.5 or

equivalent

MP3 implies MP2 but not MP1

14 (l0 for ship) = 231 (m) or (l for cargo) = 182 (m) ✓

MP3 is cao and can be awarded if MP1 is

withheld for missing squares in working but

answer is correct.

Allow ecf for MP4 for missing squares or

Yes AND calculation error provided a cargo length is

compared to a ship length based on an

either 231 > 210 OR 182 < 200 ✓ attempted relativistic calculation.

04.3 c Speed of light is the same regardless of the 1 AO2

speed of the observer or source

Total 9

How to answer it

Special Relativity: Time Dilation, Length Contraction & Invariance of c

WHAT THIS QUESTION TESTS

This question assesses core concepts from Einstein's Theory of Special Relativity:

  • Proper Time (t₀) & Time Dilation: Distinguishing proper time (measured in the rest frame of the event/object) from dilated time (measured by an observer in relative motion).
  • Proper Length (l₀) & Length Contraction: Applying length contraction along the direction of motion and comparing dimensions consistently within the same reference frame.
  • Speed of Light Invariance: Einstein's second postulate that the speed of light in a vacuum, c, is identical in all inertial reference frames, completely independent of the motion of the source or receiver.
QUESTION 04.1 • 4 MARKS

Part 1: Delivery Time & Relativistic Time Dilation

Determine whether spaceship A can deliver the food cargo within 10 days of food-experienced time.

💡 Key Knowledge

  • Lorentz Factor (γ): γ = 1 / √(1 − v²/c²)
  • When v = 0.50c :
    √(1 − v²/c²) = √(1 − 0.50²) = √(0.75) ≈ 0.8660
  • Proper Time (t₀): The time experienced by the cargo itself on spaceship A is proper time ( t₀ ≤ 10 days ).
  • Observer's Frame (Planet P): The distance measured by observer P is s = 1.4 × 10¹⁴ m at relative speed v = 0.50c = 1.50 × 10⁸ m s⁻¹ .

🧠 Exam Technique: Reference Frame Consistency

Examiners award credit for solving either completely in the spaceship's frame or completely in the planet's frame.

  • Approach A (Spaceship Frame): Calculate the actual time experienced on board ( t₀ ) and compare with 10 days.
  • Approach B (Planet Frame): Calculate the maximum allowed travel time on planet P ( t ) corresponding to t₀ = 10 days , and compare it with the actual time taken according to P.

📐 Step-by-Step Calculation (Method A: Spaceship Frame)

  1. 1 Calculate actual journey time in Planet P's frame (t):
    t = s / v = (1.4 × 10¹⁴ m) / (0.50 × 3.00 × 10⁸ m s⁻¹) = 9.333 × 10⁵ s
    In days: 9.333 × 10⁵ / (24 × 3600) ≈ 10.8 days
  2. 2 Calculate proper time experienced by cargo (t₀):
    Using time dilation: t = t₀ / √(1 − v²/c²) → t₀ = t × √(1 − 0.50²)
    t₀ = 9.333 × 10⁵ s × 0.8660 = 8.08 × 10⁵ s
    In days: 8.08 × 10⁵ / 86400 ≈ 9.35 days (accept 9.3 to 9.4 days).
  3. 3 Compare values and conclude:
    9.35 days < 10 days (or 8.08 × 10⁵ s < 8.64 × 10⁵ s ).
    Therefore, Yes, spaceship A is able to deliver the cargo within the time limit.

❌ Common Pitfalls

  • Mixing reference frames: Using the dilated time t = 10.8 days directly to claim it arrives too late. The food only ages according to proper time t₀ !
  • Incorrect time dilation formula: Writing t₀ = t / √(1 − v²/c²) (moving clocks run slow, so t > t₀ ).
  • Unit inconsistency: Forgetting to convert days into seconds ( 1 day = 86,400 s ) when using v = s/t with speeds in m s⁻¹ .
Mark Scheme Breakdown (4 Marks):
• MP1: Correct substitution of values into relativistic equation t = t₀ / √(1 − v²/c²) or length contraction equivalent.
• MP2: Correct substitution into v = s/t using values from the same frame.
• MP3: Correct intermediate calculated value seen (e.g. t₀ = 9.3 d or 9.4 d , or planet's allowed time 11.5 d ).
• MP4: Valid numerical comparison and consistent conclusion (e.g., "Yes, as 9.4 days < 10 days").
QUESTION 04.2 • 4 MARKS

Part 2: Relativistic Length Contraction & Cargo Fit

Determine whether the cargo (rest length 210 m) will fit into the cargo hold of spaceship A (measured as 200 m when moving at 0.50c).

💡 Key Knowledge

  • Length Contraction Formula: l = l₀√(1 − v²/c²)
  • Proper Length (l₀): The length measured by an observer at rest relative to the object.
  • For spaceship A: l = 200 m is its contracted length observed from P. Its rest length is l₀(ship) .
  • For cargo: l₀ = 210 m is its proper length measured at rest on planet P.

🧠 Exam Technique: Compare in the SAME Frame

You cannot compare a length in one frame to a length in another! Choose one common frame:

Frame of Comparison Spaceship Dimension Cargo Dimension Comparison
Rest Frame of Ship l₀ = 231 m l₀ = 210 m 231 m > 210 m → Fits!
Moving Frame (0.50c) l = 200 m l = 182 m 182 m < 200 m → Fits!

📐 Step-by-Step Calculation

  1. 1 Calculate relativistic factor:
    √(1 − v²/c²) = √(1 − 0.50²) = √0.75 ≈ 0.8660
  2. 2 Find the proper length of spaceship A:
    Since spaceship A is moving relative to P, its measured length is contracted: l = l₀√(1 − v²/c²)
    200 = l₀ × 0.8660 → l₀ = 200 / 0.8660 = 230.9 m ≈ 231 m
  3. 3 Compare both lengths in the ship's rest frame:
    When loaded into the ship, both are at rest relative to each other.
    Spaceship hold length = 231 m .
    Cargo length = 210 m .
    Since 231 m > 210 m (or contracted cargo 182 m < 200 m ):
    Yes, the cargo will fit into the hold.

❌ Common Pitfalls

  • Directly comparing given numbers: Concluding "No, because 210 m > 200 m". This ignores length contraction completely.
  • Contracting the ship length further: Multiplying 200 × 0.866 = 173 m instead of dividing. Remember: proper length is always the maximum measured length!
Mark Scheme Breakdown (4 Marks):
• MP1: Use of √(1 − 0.50²) or √0.75 (0.866).
• MP2: Correctly identifying l = 200 m or l₀ = 210 m .
• MP3: Correctly obtaining l₀(ship) = 231 m OR l(cargo) = 182 m .
• MP4: "Yes" AND a correct comparison: 231 > 210 OR 182 < 200 .
QUESTION 04.3 • 1 MARK

Part 3: Invariance of the Speed of Light

What is the speed of the radio signal measured by A?

✅ Correct Answer

c

Tick the box corresponding to c.

💡 Einstein's Second Postulate

Radio waves are electromagnetic radiation and travel at the speed of light in vacuum ( c ). According to Einstein’s postulate of special relativity, the speed of light in free space has the same value c for all inertial observers, regardless of the motion of the source (B) or the observer (A).

❌ Common Misconception

Students often apply Newtonian velocity addition: c + 0.50c + 0.30c = 1.8c or c + 0.50c = 1.5c . The speed of light is invariant; it is never added or subtracted from relative velocities. It is always strictly c.

Mark Scheme Breakdown (1 Mark):
• MP1: c ticked.

Topics

Optional topics · 3.12 Turning points in physics (A-level only)

Question and mark scheme from the AQA A-Level Physics examination, Paper 3 (3BD), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.