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Question 23M.3.HL.TZ1.4

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Date May 2023 Marks available [Maximum mark: 9] Reference code 23M.3.HL.TZ1.4
Level HL Paper 3 Time zone TZ1
Command term Determine, Draw, Label, Show, Show that Question number 4 Adapted from N/A
4.
[Maximum mark: 9]
23M.3.HL.TZ1.4

The diagram shows the axes of a coordinate system S at rest relative to the Earth. Earth is at the origin of S.

x' and ct′ are the coordinates of a reference frame S′ in which a spacecraft is at rest. When the origins of the two sets of axes coincide, all clocks in the frames show zero.

(a)

Show that the speed of the spacecraft is 0.80c as measured in S.

[1]

Markscheme

moves 4 ly in 5 years OR slope of angle with time axis is 0.8 ✓

 

Allow evidence for mark on the graph.

Examiners report

Spacetime diagram and two reference frames. This topic seemed to be difficult for average students.

(b.i)

An event has coordinates x = 0 and ct = 0.60 ly in S. Show, using a Lorentz transformation, that the time coordinate of this event in S′ is ct′ = 1.00 ly .

[2]

Markscheme

γ = 1.67  OR  53  OR  11-0.82 ✓

ct' = «γ(ct − vxc)» = 53 × (0.60 +0) ✓

«= 1.00 ly»

 

For MP2, working should be seen.

(b.ii)

Label, on the diagram with the letter P, the point on the ct′ axis whose ct′ coordinate is 1.00 ly.

[2]

Markscheme

identifies point with coordinates x = 0, ct = 0.60 on vertical axis ✓

draws line parallel to the x prime axis until it intersects the prime ct axis ✓

 

Award [2] for correct position of P without working shown.

(c)

At ct = 0 , a light beam is sent from Earth to a space station at rest 4.0 ly away in S. Event R is the arrival of the light beam at the space station.

Examiners report

In c), only the strongest students could determine the space coordinate of the event using the diagram and some others by calculation.

(c.i)

Draw lines to indicate R on the diagram.

[2]

Markscheme

R located at (4,4) ✓

«as intersection of» vertical line through 4 ly and photon worldline at 45 degrees✓

 

Allow MP2 even if one of the lines is not drawn.

(c.ii)

Determine, using the diagram or otherwise, the space coordinate x′ of event R in S′.

[2]

Markscheme

ALTERNATIVE 1

Using diagram:

line from R parallel to prime ct axis until it intersects space axis ✓

use of scale from (b) to estimate coordinate to x' = (1.3 ± 0.2) ly ✓

 

ALTERNATIVE 2

Using Lorentz transformation:

event R has coordinates x = ct = 4.00 ly in S ✓

so x' = «y(x − vt) = 53 × (4.00 − 0.80 × 4.00)» = 1.33 ly ✓

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