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Date	May 2017	Marks available	2	Reference code	17M.3.SL.TZ2.5
Level	Standard level	Paper	Paper 3	Time zone	2
Command term	Calculate	Question number	5	Adapted from	N/A

Question

A rocket of proper length 450 m is approaching a space station whose proper length is 9.0 km. The speed of the rocket relative to the space station is 0.80c.

X is an observer at rest in the space station.

Two lamps at opposite ends of the space station turn on at the same time according to X. Using a Lorentz transformation, determine, according to an observer at rest in the rocket,

The rocket carries a different lamp. Event 1 is the flash of the rocket’s lamp occurring at the origin of both reference frames. Event 2 is the flash of the rocket’s lamp at time ct' = 1.0 m according to the rocket. The coordinates for event 2 for observers in the space station are x and ct.

M17/4/PHYSI/SP3/ENG/TZ2/05c

Calculate the length of the rocket according to X.

[2]

a.i.

A space shuttle is released from the rocket. The shuttle moves with speed 0.20c to the right according to X. Calculate the velocity of the shuttle relative to the rocket.

[2]

a.ii.

the time interval between the lamps turning on.

[2]

b.i.

which lamp turns on first.

[1]

b.ii.

On the diagram label the coordinates x and ct.

[2]

c.i.

State and explain whether the ct coordinate in (c)(i) is less than, equal to or greater than 1.0 m.

[2]

c.ii.

Calculate the value of c ²t ² – x ².

[2]

c.iii.

Markscheme

the gamma factor is $\frac{5}{3}$ or 1.67

L = $\frac{{450}}{{\frac{5}{3}}}$ = 270 «m»

Allow ECF from MP1 to MP2.

[2 marks]

a.i.

u' = « $\frac{{u - v}}{{1 - \frac{{uv}}{{{c^2}}}}} =$ » $\frac{{0.20c - 0.80c}}{{1 - 0.20 \times 0.80}}$
OR
0.2c = $= \frac{{0.80c + u'}}{{1 + 0.80u'}}$

u' = «–»0.71c

Check signs and values carefully.

[2 marks]

a.ii.

Δt' = « $\gamma \left( {\Delta t - \frac{{v\Delta x}}{{{c^2}}}} \right) =$ » $\frac{5}{3} \times \left( {0 - \frac{{\left( {0.80c \times 9000} \right)}}{{{c^2}}}} \right)$

Δt' = «–»4.0 x 10^–5 «s»

Allow ECF for use of wrong $\gamma$ from (a)(i).

[2 marks]

b.i.

lamp 2 turns on first

Ignore any explanation

[1 mark]

b.ii.

x coordinate as shown

ct coordinate as shown

Labels must be clear and unambiguous.

Construction lines are optional.

[2 marks]

c.i.

«in any other frame» ct is greater

the interval ct' = 1.0 «m» is proper time
OR
ct is a dilated time
OR
ct = $\gamma$ ct' «= $\gamma$ »

MP1 is a statement

MP2 is an explanation

[2 marks]

c.ii.

use of c ²t ² – x ²= c ²t' ² – x' ²

c ²t ² – x ²= 1² – 0²= 1 «m²»

for MP1 equation must be used.

Award [2] for correct answer that first finds x (1.33 m) and ct (1.66 m)

[2 marks]

c.iii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

c.iii.

Syllabus sections

Option A: Relativity » Option A: Relativity (Core topics) » A.2 – Lorentz transformations

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