Explain what is meant by the statement that the spacetime interval is an invariant quantity.

calculate the spacetime interval.

determine the time between them according to observer B.

Outline why the observed times are different for A and B.

quantity that is the same/constant in all inertial frames

[1 mark]

spacetime interval = 27² – 15² = 504 «m²»

[1 mark]

ALTERNATIVE 1

Evidence of x′ = 0

t′ «\(\frac{{\sqrt {504} }}{c}\)» = 7.5 × 10^–8 «s»

ALTERNATIVE 2

γ = 1.2

t′ «= \(\frac{{9 \times {{10}^{ - 8}}}}{{1.2}}\)» = 7.5 × 10^–8 «s»

[2 marks]

observer B measures the proper time and this is the shortest time measured

OR

time dilation occurs «for B's journey» according to A

OR

observer B is stationary relative to the particle, observer A is not

[1 mark]

Calculate, for the reference frame of rocket A, the speed of rocket B according to the Galilean transformation.

Calculate, for the reference frame of rocket A, the speed of rocket B according to the Lorentz transformation.

Outline, with reference to special relativity, which of your calculations in (a) is more likely to be valid.

1.25c

[1 mark]

ALTERNATIVE 1

\(u' = \frac{{(0.50 + 0.75)}}{{1 + 0.5 \times 0.75}}c\)

0.91c

ALTERNATIVE 2

\(u' = \frac{{ - 0.50 - 0.75}}{{1 - ( - 0.5 \times 0.75)}}c\)

–0.91c

[2 marks]

nothing can travel faster than the speed of light (therefore (a)(ii) is the valid answer)

OWTTE

[1 mark]

State whether the field around the wire according to observer X is electric, magnetic or a combination of both.

Discuss the change in d according to observer Y.

Deduce whether the overall field around the wire is electric, magnetic or a combination of both according to observer Y.

magnetic field

[1 mark]

«according to Y» the positive charges are moving «to the right»

d decreases

For MP1, movement of positive charges must be mentioned explicitly.

[2 marks]

positive charges are moving, so there is a magnetic field

the density of positive charges is higher than that of negative charges, so there is an electric field

The reason must be given for each point to be awarded.

[2 marks]

Calculate the time taken for the journey in the reference frame of twin A as measured on Earth.

Determine the time taken for the journey in the reference frame of twin B.

Draw, for the reference frame of twin A, a spacetime diagram that represents the worldlines for both twins.

Suggest how the twin paradox arises and how it is resolved.

«0.6 ct = 6 ly» so t = 10 «years»

 Accept: If the 6 ly are considered to be measured from B, then the answer is 12.5 years.

ALTERNATIVE 1

10² − 6² = t² − 0²

so t is 8 «years»
Accept: If the 6 ly are considered to be measured from B, then the answer is 10 years.

ALTERNATIVE 2

gamma is \(\frac{5}{4}\)

10 × \(\frac{4}{5}\) = 8 «years»

Allow ECF from a
Allow ECF for incorrect γ in mp1

three world lines as shown

Award mark only if axes OR world lines are labelled.

according to both twins, it is the other one who is moving fast therefore clock should run slow

Allow explanation in terms of spacetime diagram.

«it is not considered a paradox as» twin B is not always in the same inertial frame of reference

OR

twin B is actually accelerating «and decelerating»

Calculate, according to Galilean relativity, the time taken for a muon to travel to the ground.

Deduce why only a small fraction of the total number of muons created is expected to be detected at ground level according to Galilean relativity.

Calculate, according to the theory of special relativity, the time taken for a muon to reach the ground in the reference frame of the muon.

Discuss how your result in (b)(i) and the outcome of the muon decay experiment support the theory of special relativity.

«\(\frac{{{{10}^4}}}{{0.995 \times 3 \times {{10}^8}}} = \)» 34 «μs»

Do not accept 10⁴/c = 33 μs.

[1 mark]

time is much longer than 10 times the average life time «so only a small proportion would not decay»

[1 mark]

\(\gamma  = 10\)

\(\Delta {t_0} = \) «\(\frac{{\Delta t}}{\gamma } = \frac{{34}}{{10}} = \)» 3.4 «μs»

[2 marks]

the value found in (b)(i) is of similar magnitude to average life time

significant number of muons are observed on the ground

«therefore this supports the special theory»

[2 marks]

State and explain the order of arrival of X and Y at the mirrors according to Jaime.

Outline whether the return of X and Y to Daniela are simultaneous according to Jaime.

beam X will reach the mirror first;

the speed of light of each beam is constant for all inertial observers;

the left mirror moves towards the beam X while the right mirror moves away from the beam Y;

the beams returning to Daniela occur at one point in space;

if this is simultaneous to Daniela, the event will also be simultaneous to Jaime;

or

beam X has less to go to the mirror and then longer to Daniela, whilst beam Y has longer to the mirror and less to Daniela;

the sum of the times are the same because Daniela is in the middle so they arrive at the same time;

Usually candidates gave acceptable answers to (a) but forgot that the speed of light must be constant for their statements to be true.

Very few answered (b) well.

State what is meant by a reference frame.

State and explain whether the force experienced by P is magnetic, electric or both, in reference frame S.

State and explain whether the force experienced by P is magnetic, electric or both, in the rest frame of P.

a set of coordinate axes and clocks used to measure the position «in space/time of an object at a particular time»
OR
a coordinate system to measure x,y,z, and t / OWTTE

[1 mark]

magnetic only

there is a current but no «net» charge «in the wire»

[2 marks]

electric only

P is stationary so experiences no magnetic force

relativistic contraction will increase the density of protons in the wire

[3 marks]

On the diagram, sketch the path of the light pulse between \({{\text{M}}_{\text{1}}}\) and \({{\text{M}}_{\text{2}}}\) as observed by Jill.

(i)     State, according to Jill, the distance moved by the spaceship in time \(\Delta t\).

(ii)     Using a Galilean transformation, derive an expression for the length of the path of the light between \({{\text{M}}_{\text{2}}}\) and \({{\text{M}}_{\text{1}}}\).

State, according to special relativity, the length of the path of the light between \({{\text{M}}_{\text{1}}}\) and \({{\text{M}}_{\text{1}}}\) as measured by Jill in terms of \(c\) and \(\Delta t\).

The time for the pulse to travel from \({{\text{M}}_{\text{2}}}\) to \({{\text{M}}_{\text{1}}}\) as measured by Ben is \(\Delta t'\). Use your answer to (b)(i) and (c) to derive a relationship between \(\Delta t\) and \(\Delta t'\).

According to a clock at rest with respect to Jill, a clock in the spaceship runs slow by a factor of 2.3. Show that the speed \(v\) of the spaceship is 0.90c.

any diagonal line as shown;

(i)     \(v\Delta t\);

(ii)     speed of pulse \({({c^2} + {v^2})^{\frac{1}{2}}}\);

distance \( = {({c^2} + {v^2})^{\frac{1}{2}}}\Delta t\);

Award [2] for bald correct answer.

\(c\Delta t\);

\(d = c\Delta t'\);

from Pythagoras \({d^2} = {c^2}\Delta {t'^2} = {c^2}\Delta {t^2} - {v^2}\Delta {t^2}\);

\(\Delta t = \frac{{\Delta t'}}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}\);

recognize that \(2.3 = \frac{1}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}\);

some evidence of rearranging e.g. \(v = \sqrt {\frac{{{{[2.3]}^2} - 1}}{{{{[2.3]}^2}}}} \);

\( = 0.90{\text{c}}\)

Most candidates were able to draw the correct path of the light.

Parts (b), (c) and (d) effectively dealt with the derivation of the time dilation formula and here many candidates had problems often relying on guesswork and half-remembered proofs rather than follow the logical development of the questions. The calculation was often done well but with the usual confusion between the times.

Parts (b), (c) and (d) effectively dealt with the derivation of the time dilation formula and here many candidates had problems often relying on guesswork and half-remembered proofs rather than follow the logical development of the questions. The calculation was often done well but with the usual confusion between the times.

Parts (b), (c) and (d) effectively dealt with the derivation of the time dilation formula and here many candidates had problems often relying on guesswork and half-remembered proofs rather than follow the logical development of the questions. The calculation was often done well but with the usual confusion between the times.

While moving away from the base station, Suzanne observes another spacecraft travelling towards her at a speed of 0.8c. Using Galilean transformations, calculate the relative speed of the two spacecraft.

Using the postulates of special relativity, state and explain why Galilean transformations cannot be used in this case to find the relative speeds of the two spacecraft.

Using relativistic kinematics, the relative speeds of the two spacecraft is shown to be 0.976c. Suzanne measures the other spacecraft to have a length of 8.00 m. Calculate the proper length of the other spacecraft.

Suzanne’s spacecraft is on a journey to a star. According to Juan, the distance from the base station to the star is 11.4 ly. Show that Suzanne measures the time taken for her to travel from the base station to the star to be about 9 years.

1.6c;

(one of the) postulates states that the speed of light in a vacuum is the same for all inertial observers;

Galilean transformation will give a relative speed greater than the speed of light;

\(\gamma  = \frac{1}{{\sqrt {1 - {{0.976}^2}} }}{\text{ }}( = 4.59)\);

\({l_0} = (4.56 \times 8.00 = ){\text{ 36.7 (m)}}\);

Note: the final answer for SP3 is different to the HP3.

\(t = \frac{s}{v} = \frac{{11.4}}{{0.8}} = 14.25{\text{ (years)}}\);

\(\Delta {t_0} = \frac{{\Delta t}}{\gamma } = \frac{{14.25}}{{1.67}} = 8.6{\text{ (years)}}\);

Allow ECF from (b).

Accept length contraction with the same result.

Only HL Questions 12(a), (b)(i) and (c) were common with SL questions 12(a), (b)(i) and (c). Many did not address “frame of reference”, only explaining “inertial”. Most could identify the postulate relevant to Galilean transformations but few could earn full marks. The calculation was well done by those who attempted the question.

Only HL Questions 12(a), (b)(i) and (c) were common with SL questions 12(a), (b)(i) and (c). Many did not address “frame of reference”, only explaining “inertial”. Most could identify the postulate relevant to Galilean transformations but few could earn full marks. The calculation was well done by those who attempted the question.

Only HL Questions 12(a), (b)(i) and (c) were common with SL questions 12(a), (b)(i) and (c). Many did not address “frame of reference”, only explaining “inertial”. Most could identify the postulate relevant to Galilean transformations but few could earn full marks. The calculation was well done by those who attempted the question.

Only HL Questions 12(a), (b)(i) and (c) were common with SL questions 12(a), (b)(i) and (c). Many did not address “frame of reference”, only explaining “inertial”. Most could identify the postulate relevant to Galilean transformations but few could earn full marks. The calculation was well done by those who attempted the question.

Only HL Questions 12(a), (b)(i) and (c) were common with SL questions 12(a), (b)(i) and (c). Many did not address “frame of reference”, only explaining “inertial”. Most could identify the postulate relevant to Galilean transformations but few could earn full marks. The calculation was well done by those who attempted the question.

Only HL Questions 12(a), (b)(i) and (c) were common with SL questions 12(a), (b)(i) and (c). Many did not address “frame of reference”, only explaining “inertial”. Most could identify the postulate relevant to Galilean transformations but few could earn full marks. The calculation was well done by those who attempted the question.

Muons are unstable particles with a proper lifetime of 2.2 μs. Muons are produced 2.0 km above ground and move downwards at a speed of 0.98c relative to the ground. For this speed \(\gamma \) = 5.0. Discuss, with suitable calculations, how this experiment provides evidence for time dilation.

ALTERNATIVE 1 — for answers in terms of time
overall idea that more muons are detected at the ground than expected «without time dilation»

«Earth frame transit time = \(\frac{{2000}}{{0.98c}}\)» = 6.8 «μs»

«Earth frame dilation of proper half-life = 2.2 μs x 5» = 11 «μs»
OR
«muon’s proper transit time = \(\frac{{6.8\mu s}}{5}\)» = 1.4 «μs»

ALTERNATIVE 2 – for answers in terms of distance
overall idea that more muons are detected at the ground than expected «without time dilation»

«distance muons can travel in a proper lifetime = 2.2 μs x 0.98c» = 650 «m»

«Earth frame lifetime distance due to time dilation = 650 m x 5» = 3250 «m»
OR
«muon frame distance travelled = \(\frac{{2000}}{5}\)» = 400 «m»

Accept answers from one of the alternatives.

[3 marks]

For the reference frame of the electron, calculate the distance travelled by the detector.

For the reference frame of the laboratory, calculate the time taken for the electron to reach the detector after its emission from the nucleus.

For the reference frame of the electron, calculate the time between its emission at the nucleus and its detection.

Outline why the answer to (c) represents a proper time interval.

γ=4.503

\( \ll \frac{{0.800}}{{4.50}} =  \gg 0.178{\rm{m}}\)

\({\rm{time}} = \frac{{0.800}}{{2.94 \times {{10}^8}}}\)

2.74 ns

\(\frac{{2.74}}{{4.5}}\) OR \(\frac{{0.178}}{{2.94 \times {{10}^8}}}\)

0.608 ns

it is measured in the frame of reference in which both events occur at the same position
OR
it is the shortest time interval possible

Define proper length.

A charged pion decays spontaneously in a time of 26 ns as measured in the frame of reference in which it is stationary. The pion moves with a velocity of 0.96c relative to the Earth. Calculate the pion’s lifetime as measured by an observer on the Earth.

In the pion reference frame, the Earth moves a distance X before the pion decays. In the Earth reference frame, the pion moves a distance Y before the pion decays. Demonstrate, with calculations, how length contraction applies to this situation.

the length of an object in its rest frame

\(\frac{1}{{\sqrt {\left( {1 - {{0.96}^2}} \right)} }}\) OR \(\gamma  = 3.6\)
ECF for wrong \(\gamma\)

93 «ns»
Award [2] for a bald correct answer.

«X is» 7.5 «m» in frame on pion

«Y is» 26.8 «m» in frame on Earth 

identifies proper length as the Earth measurement
OR
identifies Earth distance according to pion as contracted length
OR
a statement explaining that one of the length is shorter than the other one

Outline the conclusion from Maxwell’s work on electromagnetism that led to one of the postulates of special relativity.

light is an EM wave

speed of light is independent of the source/observer

State what is meant by inertial in this context.

An observer is travelling at velocity v towards a light source. Determine the value the observer would measure for the speed of light emitted by the source according to

(i) Maxwell’s theory.

(ii) Galilean transformation.

not being accelerated
OR
not subject to an unbalanced force
OR
where Newton’s laws apply

(i) c

(ii) c+v

State the reason why the time interval between event 1 and event 2 is a proper time interval as measured by Judy.

(i)     Calculate the time interval between event 1 and event 2 according to Peter.

(ii)     Calculate the time interval between event 1 and event 2 according to Judy.

(i)     Calculate, according to Judy, the distance separating the Earth and planet P.

(ii)     Using your answers to (b)(ii) and (c)(i), determine the speed of planet P relative to the spaceship.

(iii)     Comment on your answer to (c)(ii).

Determine, according to Judy in the spaceship, which signal is emitted first.

because the events occur at the same place/point in space for this observer;

Do not allow “events within the same reference frame”.

(i)     \(t = \left( {\frac{{12}}{{0.60{\text{c}}}} = } \right){\text{ 20(yr)}}\);

(ii)     \(\gamma  = \left( {\frac{1}{{\sqrt {1 - {{0.60}^2}} }} = } \right){\text{ 1.25}}\); (allow implicit value)

\({t_{{\text{rocket}}}} = \left( {\frac{{20 {\text{yr}}}}{\gamma } = } \right){\text{ 16(yr)}}\); (allow ECF)

Award [2] for a bald correct answer.

(i)     \(L = \left( {\frac{{12{\text{ly}}}}{\gamma } = } \right){\text{ 9.6(ly)}}\); (allow ECF from (b)(ii))

(ii)     \(v = \left( {\frac{{9.6{\text{ly}}}}{{16{\text{y}}}} = } \right){\text{ 0.60c}}\); (allow ECF from (b)(ii) and (c)(i))

(iii)     (by principle of relativity this should be the) same as the speed of the spaceship relative to Earth;

both signals travel at the same speed c;

Judy must agree that the signals arrive at S simultaneously / OWTTE;

for Judy, observer S moves away from the signal traveling from P/towards the signal traveling from Earth;

for Judy the signal from P has further to travel to reach S – so was emitted first;

Do not accept explanations based on Judy approaching P or seeing/receiving the signal from P first as this is irrelevant.

Award [0] for a bald correct answer.

One of the postulates of special relativity refers to the speed of light. State the other postulate of special relativity.

the laws of physics are the same for all inertial observers;

(a) was well answered.

An observer at rest relative to Earth observes two spaceships. Each spaceship is moving with a speed of 0.85 c but in opposite directions. The observer measures the rate of increase of distance between the spaceships to be 1.7 c. Outline whether this observation contravenes the theory of special relativity.

The observer on Earth in (a) watches one spaceship as it travels to a distant star at a speed of 0.85 c. According to observers on the spaceship, this journey takes 8.0 years.

(i) Calculate, according to the observer on Earth, the time taken for the journey to the star.

(ii) Calculate, according to the observer on Earth, the distance from Earth to the star.

(iii) At the instant when the spaceship passes the star, the observer on the spaceship sends a radio message to Earth. The spaceship continues to move at a speed of 0.85 c. Determine, according to the spaceship observer, the time taken for the message to arrive on Earth.

State one prediction of Maxwell’s theory of electromagnetism that is consistent with special relativity.

A current is established in a long straight wire that is at rest in a laboratory.

A proton is at rest relative to the laboratory and the wire.

Observer X is at rest in the laboratory. Observer Y moves to the right with constant speed relative to the laboratory. Compare and contrast how observer X and observer Y account for any non-gravitational forces on the proton.

the speed of light is a universal constant/invariant
OR
c does not depend on velocity of source/observer

electric and magnetic fields/forces unified/frame of reference dependant

[1 mark]

observer X will measure zero «magnetic or electric» force

observer Y must measure both electric and magnetic forces

which must be equal and opposite so that observer Y also measures zero force

Allow [2 max] for a comment that both X and Y measure zero resultant force even if no valid explanation is given.

[3 marks]

Two protons are moving with the same velocity in a particle accelerator.

Observer X is at rest relative to the accelerator. Observer Y is at rest relative to the protons.

Explain the nature of the force between the protons as observed by observer X and observer Y.

Y measures electrostatic repulsion only
protons are moving relative to X «but not Y» OR protons are stationary relative to Y
moving protons create magnetic fields around them according to X
X also measures an attractive magnetic force OR relativistic/Lorentz effects also present

State the postulate of special relativity that is related to the speed of light.

A rocket moving at a relativistic speed passes an observer who is at rest on the ground equidistant from two trees L and R. At the moment that an observer in the rocket is opposite the ground observer, lightning strikes L and R at the same time according to the
ground observer. Light from the strikes reaches the observer in the rocket as well as the observer on the ground.

(i) Explain why, according to the observer in the rocket, light from the two strikes will reach the ground observer at the same time.

(ii) Using your answer to (a) and (b)(i), outline why, according to the rocket observer, tree R was hit by lightning before tree L.

the speed of light in a vacuum is the same for all inertial observers/observers in uniform motion;

(i) ground observer measures a zero proper time interval for the two arrivals;
all other observers measure a time interval of γ×0=0;
hence the arrivals are simultaneous for all observers, including rocket 
Award [1] for statement that “events that are simultaneous for one observer
and occur at the same place are simultaneous for all observers”.

(ii) according to the rocket observer, the ground observer moves towards the signal from tree L and away from the signal from tree R;
since the signals move at the same speed and they arrive at the same time according to the rocket observer;
signal from tree R must have been emitted first

Explain what is meant by proper length.

Draw a spacetime diagram for this situation according to an observer at rest relative to the tunnel.

Calculate the velocity of the train, according to an observer at rest relative to the tunnel, at which the train fits the tunnel.

For an observer on the train, it is the tunnel that is moving and therefore will appear length contracted. This seems to contradict the observation made by the observer at rest to the tunnel, creating a paradox. Explain how this paradox is resolved. You may refer to your spacetime diagram in (b).

the length of an object in its rest frame

OR

the length of an object measured when at rest relative to the observer

world lines for front and back of tunnel parallel to ct axis

world lines for front and back of train

which are parallel to ct′ axis

realizes that gamma = 1.25

0.6c

ALTERNATIVE 1

indicates the two simultaneous events for t frame

marks on the diagram the different times «for both spacetime points» on the ct′ axis «shown as Δt′ on each diagram»

ALTERNATIVE 2: (no diagram reference)

the two events occur at different points in space

statement that the two events are not simultaneous in the t′ frame

Two space stations X and Y are at rest relative to each other. The separation of X and Y as measured in their frame of reference is 1.80×10¹¹m.

State what is meant by a frame of reference.

A radio signal is sent to both space stations in (a) from a point midway between them. On receipt of the signal a clock in X and a clock in Y are each set to read zero. A spaceship S travels between X and Y at a speed of 0.750c as measured by X and Y. In the frame of reference of S, station X passes S at the instant that X’s clock is set to zero. A clock in S is also set to zero at this instant.

(i) Calculate the time interval, as measured by the clock in X, that it takes S to travel from X to Y.

(ii) Calculate the time interval, as measured by the clock in S, that it takes S to travel from X to Y.

(iii) Explain whether the clock in X or the clock in S measures the proper time.

(iv) Explain why, according to S, the setting of the clock in X and the setting of the clock in Y does not occur simultaneously.

a set of coordinates that can be used to locate events/position of objects;

(i) \(\frac{{1.80 \times {{10}^{11}}}}{{0.750 \times 3 \times {{10}^8}}}\);
=800(s);
Award [2] for a bald correct answer.

(ii) \(\gamma  = \left( {\frac{1}{{\sqrt {1 - {{0.750}^2}} }} = } \right)1.51\);
\({\rm{time}} = \left( {\frac{{800}}{{1.51}} = } \right)530\left( {\rm{s}} \right)\);

Watch for ECF from (b)(i) or first marking point in (b)(ii).
Award [2] for a bald correct answer.

(iii) only S’s clock measures proper time;
because S’s clock is at both events / events occur at same place in S’s frame;

(iv) according to S, Y moves towards/X moves away from the radio signal;
the signal travels at the same speed/at the speed of light in each direction;
therefore according to S’s clock the signal reaches Y before it reaches X/X after reaching Y;

or

S’s frame is different/moving relative to the X and Y frame;
the two events/arrival of signals are separated in space;
so if simultaneous for XY, cannot be simultaneous for S;

Define frame of reference.

Calculate, according to the observer on Earth, the time taken for A and B to meet.

Identify the terms in the formula.

u′ = \(\frac{{u - v}}{{1 - \frac{{uv}}{{{c^2}}}}}\)

Determine, according to an observer in A, the velocity of B.

Determine, according to an observer in A, the time taken for B to meet A.

Deduce, without further calculation, how the time taken for A to meet B, according to an observer in B, compares with the time taken for the same event according to an observer in A.

a co-ordinate system in which measurements «of distance and time» can be made

Ignore any mention to inertial reference frame.

closing speed = c

2 «s»

u and v are velocities with respect to the same frame of reference/Earth AND u′ the relative velocity

Accept 0.4c and 0.6c for u and v

\(\frac{{ - 0.4 - 0.6}}{{1 + 0.24}}\)

«–» 0.81c

\(\gamma \) = 1.25

so the time is t = 1.6 «s»

gamma is smaller for B

so time is greater than for A

Carrie measures her spaceship to have a length of 100m. Peter measures Carrie’s spaceship to have a length of 91m.

(i) Explain why Carrie measures the proper length of the spaceship.

(ii) Show that Carrie travels at a speed of approximately 0.4 c relative to Peter.

According to Carrie, it takes the star ten years to reach her. Using your answer to (a)(ii), calculate the distance to the star as measured by Peter.

According to Peter, as Carrie passes the star she sends a radio signal. Determine the time, as measured by Carrie, for the message to reach Peter.

(i) proper length is measured by observer at rest relative to object / Carrie is at rest relative to spaceship;

(ii) \(\gamma  = \left( {\frac{{100}}{{91}} = } \right)1.1\);
evidence of algebraic manipulation e.g. \(\frac{{{v^2}}}{{{c^2}}} = 1 - \frac{1}{{{{1.1}^2}}}\) to give v=0.42c;
≈0.4c

travel time measured by Peter = (10×γ=)11years;
4.6ly or 4.4ly (if 0.4 c used);

moves away at 0.42 c so is 4.2ly away when signal emitted; (allow ECF from (a)(ii))
signal travel time t where ct=4.2+0.42ct;
7.2y or 7y (if 0.4 c used);

In the context of the theory of relativity, state what is meant by an event.

(i) Calculate the time interval t between event 1 and event 2 according to an observer in the box.

(ii) According to an observer on the ground the time interval between event 1 and event 2 is T. One student claims that \(T = \frac{t}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}\) and another that \(T = t\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \).

Explain why both students are wrong.

Relative to an observer on the ground,

(i) calculate the distance between S and D.

(ii) state the speed of the photon leaving S.

(iii) state an expression for the distance travelled by detector D in the time interval T (T is the interval in (b)(ii)).

(iv) determine T, using your answers to (c)(i), (ii) and (iii).

a point in spacetime / something happening at a particular time and a particular point in space;

(i) \(t = \frac{{6.0}}{{3.0 \times {{10}^8}}} = 2.0 \times {10^{ - 8}}{\rm{s}}\);

(ii) for either formula to be used one of the time intervals must be a proper time interval;
the two events occur at different points in space and so neither observer measures a proper time interval;
the proper time interval is that of the photons;

(i) \(\gamma  = \frac{1}{{\sqrt {1 - {{0.80}^2}} }} = \frac{5}{3} = 1.67\);
\(l = \frac{L}{\gamma } = \frac{{6.0}}{{1.67}} = 3.6{\rm{m}}\);
Award [2] for a bald correct answer.

(ii) c;

(iii) vT or 0.80cT;

(iv) cT=0.80cT+3.6;
\(T = \frac{{3.6}}{{0.20 \times 3.0 \times {{10}^8}}} = 6.0 \times {10^{ - 8}}{\rm{s}}\);
Award [2] for a bald correct answer.

Define proper length.

A spaceship is travelling to the right at speed 0.75 c, through a tunnel which is open at both ends. Observer A is standing at the centre of one side of the tunnel. Observer A, for whom the tunnel is at rest, measures the length of the tunnel to be 240 m and the length of the spaceship to be 200 m. The diagram below shows this situation from the perspective of observer A.

Observer B, for whom the spaceship is stationary, is standing at the centre of the spaceship.

(i) Calculate the Lorentz factor, γ, for this situation.

(ii) Calculate the length of the tunnel according to observer B.

(iii) Calculate the length of the spaceship according to observer B.

(iv) According to observer A, the spaceship is completely inside the tunnel for a short time. State and explain whether or not, according to observer B, the spaceship is ever completely inside the tunnel.

Two sources of light are located at each end of the tunnel. The diagram below shows this situation from the perspective of observer A.

According to observer A, at the instant when observer B passes observer A, the two sources of light emit a flash. Observer A sees the two flashes simultaneously. Discuss whether or not observer B sees the two flashes simultaneously.

the length of an object as measured by an observer who is at rest relative to the object;

(i) \(\gamma  = \frac{1}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }} = \frac{1}{{\sqrt {1 - {{0.75}^2}} }} = 1.5\);

(ii) \(L = \frac{{{L_0}}}{\gamma } = \frac{{240}}{{1.5}} = 160{\rm{m}}\);

(iii) \({L_0} = \gamma L = 1.5 \times 200 = 300{\rm{m}}\); 

(iv) the spaceship is never completely inside the tunnel;
because (according to observer B) the spaceship is longer than the tunnel;
Apply ECF in all parts of question (b).

observer B will not see the two flashes simultaneously;
according to B, light 2 is moving to the left/towards observer B;
since the speed of light is the same for both sources;
the flash from light 2 reaches B before the flash from light 1;

or

according to B, the two flashes arrive at A simultaneously;
according to B, A is moving to the left/away from light 2;
since light from both sources moves with the same speed;
for the flashes to be received by A at the same time, the flash from light 2 must be emitted first; 
Accept any equivalent discussion.

Calculate the angle between the worldline of S and the worldline of the Earth.

Draw, on the diagram, the x′-axis for the reference frame of S.

An event Z is shown on the diagram. Label the co-ordinates of this event in the reference frame of S.

angle = tan^–1 «\(\frac{{0.8}}{1}\)» = 39 «^o» OR 0.67 «rad»

adds x′-axis as shown

Approximate same angle to v = c as for ct′.

Ignore labelling of that axis.

adds two lines parallel to ct′ and x′ as shown indicating coordinates

Draw on the spacetime diagram the worldline of B according to the Earth observer and label it B.

Deduce, showing your working on the spacetime diagram, the value of ct according to the Earth observer at which the rocket B emitted its flash of light.

Explain whether or not the arrival times of the two flashes in the Earth frame are simultaneous events in the frame of rocket A.

Calculate the velocity of rocket B relative to rocket A.

straight line with negative gradient with vertical intercept at ct = 1.2 «km»

through (–0.6, 2.2) ie gradient = –1.67

Tolerance: Allow gradient from interval –2.0 to –1.4, (at ct = 2.2, x from interval 0.5 to 0.7).

If line has positive gradient from interval 1.4 to 2.0 and intercepts at ct = 1.2 km then allow [1 max].

[2 marks]

line for the flash of light from A correctly drawn

line for the flash of light of B correctly drawn

correct reading taken for time of intersection of flash of light and path of B, ct = 2.4 «km»

Accept values in the range: 2.2 to 2.6.

[3 marks]

the two events take place in the same point in space at the same time

so all observers will observe the two events to be simultaneous / so zero difference

Award the second MP only if the first MP is awarded.

[2 marks]

\(u' = \frac{{ - 0.6 - 0.8}}{{1 - ( - 0.6) \times 0.8}}\)

= «–»0.95 «c»

[2 marks]

Calculate the length of the rocket according to X.

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.

the time interval between the lamps turning on.

which lamp turns on first.

On the diagram label the coordinates x and ct.

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

Calculate the value of c ²t ² – x ².

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]

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]

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]

lamp 2 turns on first

Ignore any explanation

[1 mark]

x coordinate as shown

ct coordinate as shown

Labels must be clear and unambiguous.

Construction lines are optional.

[2 marks]

«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]

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]

Define frame of reference.

In the reference frame of the laboratory the force on X is magnetic.

(i) State the nature of the force acting on X in this reference frame where X is at rest.

(ii) Explain how this force arises.

a coordinate system
OR
a system of clocks and measures providing time and position relative to an observer

OWTTE

i

electric
OR
electrostatic

ii

«as the positive ions are moving with respect to the charge,» there is a length contraction

therefore the charge density on ions is larger than on electrons

so net positive charge on wire attracts X

For candidates who clearly interpret the question to mean that X is now at rest in the Earth frame accept this alternative MS for bii
the magnetic force on a charge exists only if the charge is moving
an electric force on X , if stationary, only exists if it is in an electric field
no electric field exists in the Earth frame due to the wire
and look back at b i, and award mark for there is no electric or magnetic force on X

One of the two postulates of special relativity may be stated as:

“The laws of physics are the same for all observers in inertial reference frames.”

State

(i) what is meant by an inertial frame of reference.

(ii) the other postulate of special relativity.

In a thought experiment to illustrate the concept of simultaneity, Vladimir is standing on the ground close to a straight, level railway track. Natasha is in a railway carriage that is travelling along the railway track with constant speed v in the direction shown.

Natasha is sitting on a chair that is equidistant from each end of the carriage. At either end of the carriage are two clocks C₁ and C₂. Next to Natasha is a switch that, when operated, sends a signal to each clock. The clocks register the time of arrival of the signals. At the instant that Natasha and Vladimir are opposite each other, Natasha operates the switch. According to Natasha, C₁ and C₂ register the same time of arrival of each signal.

Explain, according to Vladimir, whether or not C₁ and C₂ register the same time of arrival for each signal.

The speed v of the carriage is 0.70c. Vladimir measures the length of the table at which Natasha is sitting to be 1.0 m.

(i) Calculate the length of the table as measured by Natasha.

(ii) Explain which observer measures the proper length of the table.

(i) (a reference frame) in which Newton’s first law holds true/that is not accelerating/that is moving with constant velocity;

(ii) the speed of light in a vacuum/free space is the same for all inertial observers;

Look for these main points:
signal from switch travels at same speed c to each lamp;
but during signal transfer C₁ moves closer to/C₂ moves away from source of signal;
since speed of light is independent of speed of source, signal reaches C₁ before C₂/C₂ after C₁;
according to Vladamir C₁ registers arrival of signal before C₂/C₂ registers arrival of signal after C₁;

(i) \(\gamma  = \frac{1}{{\sqrt {1 - {{\left( {0.70} \right)}^2}} }} = 1.4\);
L₀=γL;
=1.4m;

(ii) Natasha
since proper length is defined as the length of the object measured by the observer at rest with respect to the object;

Calculate the velocity of the spaceship relative to the Earth.

The spaceship passes the space station 90 minutes later as measured by the spaceship clock. Determine, for the reference frame of the Earth, the distance between the Earth and the space station.

As the spaceship passes the space station, the space station sends a radio signal back to the Earth. The reception of this signal at the Earth is event A. Determine the time on the Earth clock when event A occurs.

Construct event A and event B on the spacetime diagram.

Estimate, using the spacetime diagram, the time at which event B occurs for the spaceship.

0.60c

OR

1.8 × 10⁸ «m s^–1»

[1 mark]

ALTERNATIVE 1

time interval in the Earth frame = 90 × γ = 112.5 minutes

«in Earth frame it takes 112.5 minutes for ship to reach station»

so distance = 112.5 × 60 × 0.60c

1.2 × 10m¹² «m»

ALTERNATIVE 2

Distance travelled according in the spaceship frame = 90 × 60 × 0.6c

= 9.72 × 10¹¹ «m»

Distance in the Earth frame «= 9.72 × 10¹¹ × 1.25» = 1.2 × 10¹² «m»

[3 marks]

signal will take «112.5 × 0.60 =» 67.5 «minutes» to reach Earth «as it travels at c»

OR

signal will take «\(\frac{{1.2 \times {{10}^{12}}}}{{3 \times {{10}^8}}}\) =» 4000 «s»

total time «= 67.5 + 112.5» = 180 minutes or 3.00 h or 3:00am

[2 marks]

line from event E to A, upward and to left with A on ct axis (approx correct)

line from event A to B, upward and to right with B on ct' axis (approx correct)

both lines drawn with ruler at 45 (judge by eye)

eg:

[3 marks]

ALTERNATIVE 1

«In spaceship frame»

Finds the ratio \(\frac{{OB}}{{OE}}\) (or by similar triangles on x or ct axes), value is approximately 4

hence time elapsed ≈ 4 × 90 mins ≈ 6h «so clock time is ≈ 6:00»

Alternative 1:

Allow similar triangles using x-axis or ct-axis, such as \(\frac{{distance\,2}}{{distance\,1}}\) from diagrams below

ALTERNATIVE 2

«In Earth frame»

Finds the ratio

\(\frac{{ct{\text{ coordinate of B}}}}{{ct{\text{ coordinate of A}}}}\), value is approximately 2.5

hence time elapsed ≈ \(\frac{{2.5 \times 3{\text{h}}}}{{1.25}}\) ≈ 6h

«so clocktime is ≈ 6:00»

ALTERNATIVE 2:

[2 marks]

State what is meant by an inertial frame of reference.

A spaceship travels from space station Alpha to space station Zebra at a constant speed of 0.90c relative to the space stations. The distance from Alpha to Zebra is 10ly according to space station observers. At this speed γ=2.3.

Calculate the time taken to travel between Alpha and Zebra in the frame of reference of an observer

(i) on the Alpha space station.

(ii) on the spaceship.

Explain which of the time measurements in (b)(i) and (b)(ii) is a proper time interval.

The spaceship arrives at Zebra and enters an airlock at constant speed. O is an observer at rest relative to the airlock. Two lamps P and Q emit a flash simultaneously according to the observer S in the spaceship. At that instant, O and S are opposite each other and midway between the lamps.

Discuss whether the lamps flash simultaneously according to observer O.

a co-ordinate system (in which measurements of distance and time can be made);
which is not accelerating;
in which Newton’s laws are valid;

(i) \(\left( {\frac{{10}}{{0.90{\rm{c}}}} = } \right)11{\rm{yr}}\);
\(\left( { = 3.5 \times {{10}^8}{\rm{s}}} \right)\);
This is a question testing units for this option. Do not award mark for an incorrect or missing unit.

(ii) distance according to spaceship observer \( = \frac{{10}}{{2.3}}\left( { = 4.3{\rm{ly}}} \right)\);
so time for spaceship \( = \frac{{4.3}}{{0.90}} = 4.8\left( {{\rm{yr}}} \right)\);

between two events occurring at the same point in space / shortest time measured;
so proper time interval measured by observer on spaceship;
Do not award second marking point unless a reason has been attempted.

speed of light is the same for both observers O and S / events simultaneous in stationary reference frame are not (necessarily) simultaneous in moving reference frame;
S is moving so PS will be longer than QS when light reaches S;
so if light arrives simultaneously then light from P will have been in transit for longer than Q;
therefore P emits a flash before Q;

Describe what is meant by a frame of reference.

At a later time the police spacecraft is alongside Speedy’s spacecraft. The police spacecraft is overtaking Speedy’s spacecraft at a constant velocity.

Officer Sylvester is at a point midway between the flashing lamps, both of which he can see. At the instant when Officer Sylvester and Speedy are opposite each other, Speedy observes that the blue lamps flash simultaneously.

State and explain which lamp is observed to flash first by Officer Sylvester.

The police spacecraft is travelling at a constant speed of 0.5c relative to Speedy’s frame of reference. The light from a flash of one of the lamps travels across the police spacecraft and is reflected back to the light source. Officer Sylvester measures the time taken for
the light to return to the source as 1.2 × 10^–8s.

(i) Outline why the time interval measured by Officer Sylvester is a proper time interval.

(ii) Determine, as observed by Speedy, the time taken for the light to travel across the police spacecraft and back to its source.

a coordinate system / set of rulers / clocks;
in which measurements of distance/position and time can be made;

light travels at same speed for both observers;
during transit time Officer Sylvester moves towards point of emission at front/away from point of emission at back;
light from front arrives first as distance is less / light from back arrives later as distance is more;
Officer Sylvester observes the front lamp flashes first;

Award [0] for a bald correct answer without correct explanation

(i) the two events occur at the same place (in the same frame of reference) / shortest measured time;

(ii) \(y = \left( {\frac{1}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }} = } \right)1.15\);
Δt=1.15 x Δt₀;
1.48 x 10^-8(s);

Determine the time, in years, that it takes the spacecraft to reach the planet according to the

(i) observer on Earth.

(ii) observer in the spacecraft.

The spacecraft passes a space station that is at rest relative to the Earth. The proper length of the space station is 310 m.

(i) State what is meant by proper length.

(ii) Calculate the length of the space station according to the observer in the spacecraft.

F and B are two flashing lights located at the ends of the space station, as shown. As the spacecraft approaches the space station in (b), F and B turn on. The lights turn on simultaneously according to the observer on the space station who is midway between the lights.

State and explain which light, F or B, turns on first according to the observer in the spacecraft.

(i) \(\left( {\frac{{12{\rm{ly}}}}{{0.60{\rm{c}}}} = } \right)20\left( {{\rm{yr}}} \right)\) or 6.3×108 (s);

(ii) \(y = \left( {\frac{1}{{\sqrt {1 - {{0.60}^2}} }} = } \right)1.25\);
\(\Delta {t_0} = \left( {\frac{{\Delta t}}{y} = \frac{{20}}{{1.25}} = } \right)16\left( {{\rm{yr}}} \right)\) or 5.0×10 8 (s); (allow ECF from (a)(i));

State and explain whether the pendulum period is a proper time interval for observer T, observer G or both T and G.

Observer T measures the period of oscillations of the pendulum to be 0.850s. Calculate the period of oscillations according to observer G.

Observer T is standing in the middle of a train watched by observer G at the side of the track. Two lightning strikes hit the ends of
the train. The strikes are simultaneous according to observer T.

Light from the strikes reaches both observers.

(i) Explain why, according to observer G, light from the two strikes reaches observer T at the same time.

(ii) Using your answer to (i), explain why, according to observer G, end X of the train was hit by lightning first.

only T measures the proper time interval;
for T the pendulum is (a single clock) at rest/same point in space;
Do NOT simply allow that the pendulum is in the same frame as T.

\(\gamma  = \left( {\frac{1}{{\sqrt {1 - {{0.95}^2}} }} = } \right)3.20\);
\(T = \left( {\gamma {T_0} = 3.20 \times 0.85 = } \right)2.72{\rm{s}}\);

(i) the arrivals at T of the light from the two strikes occurs at the same point in space for T and are simultaneous for T;
so the arrivals of the light are simultaneous for all other observers as well;

or

T measures a zero proper time interval for the arrivals of the light;
so G measures a time interval equal to \(\gamma \)×0=0 also;

(ii) according to G, T is moving away from the light from the left strike and yet receives the light at the same time as the light from the right strike;
since the speed of light is the same for light from both strikes;
the left strike occurred first
No marks for just stating left strike is first. Given.

State which of the two time intervals is a proper time.

Calculate the speed v of the train for the ratio \(\frac{{\Delta {t_{\text{P}}}}}{{\Delta {t_{\text{Q}}}}} = 0.30\).

Later the train is travelling at a speed of 0.60c. Observer P measures the length of the train to be 125 m. The train enters a tunnel of length 100 m according to observer Q.

Show that the length of the train according to observer Q is 100 m.

Draw the time \(ct'\) and space \(x'\) axes for observer P’s reference frame on the spacetime diagram.

Deduce, using the spacetime diagram, which light was turned on first according to observer P.

Apply a Lorentz transformation to show that the time difference between events B and F according to observer P is 2.5 × 10^–7 s.

Demonstrate that the spacetime interval between events B and F is invariant.

A second train is moving at a velocity of –0.70c with respect to the ground.

Calculate the speed of the second train relative to observer P.

Δt_P / observer sitting in the train

[1 mark]

\(\gamma  = \frac{{\Delta {t_{\text{Q}}}}}{{\Delta {t_{\text{P}}}}} = \) «\( = \frac{1}{{0.30}}\)» = 3.3

to give v = 0.95c

[2 marks]

\(\gamma \) = 1.25

«length of train according Q» = 125/1.25

«giving 100m»

[2 marks]

axes drawn with correct gradients of \(\frac{5}{3}\) for \(ct'\) and 0.6 for \(x'\)

Award [1] for one gradient correct and another approximately correct.

[1 mark]

lines parallel to the \(x'\) axis and passing through B and F

intersections on the \(ct'\) axis at \({\text{B'}}\) and \({\text{F'}}\) shown

light at the front of the train must have been turned on first

[3 marks]

\(\Delta t' = 1.25 \times \frac{{0.6 \times 100}}{{3 \times {{10}^8}}}\)

«2.5 × 10⁻⁷»

Allow ECF for gamma from (c).

[1 mark]

according to P: \({\left( {3 \times {{10}^8} \times 2.5 \times {{10}^{ - 7}}} \right)^2} - {125^2} = \) «−» 10000

according to Q: \({\left( {3 \times {{10}^8} \times 0} \right)^2} - {100^2} = \) «−» 10000

[2 marks]

\(u' = \frac{{ - 0.7 - 0.6}}{{1 + 0.7 \times 0.6}}\) c

= «−» 0.92c

[2 marks]

For the reference frame of the Earth observer, calculate the speed of rocket A in terms of the speed of light c.

Using the graph opposite, deduce the order in which

(i) the beacons flash in the reference frame of rocket A.

(ii) the Earth observer sees the beacons flash.

Δct=2.0km AND Δx=0.8km

\(v =  \ll \frac{{\Delta x}}{{\Delta ct}} = \frac{{0.8}}{{2.0}} =  \gg 0.4c\)

Allow any correct read-off from graph.
Accept answers from 0.37c to 0.43c.

(i) events at same perpendicular distance from x′ axis of rocket are simultaneous OR line joining X to Y is parallel to x′ axis

X and Y simultaneously then Z

MP1 may be present on spacetime diagram.

(ii) constructs light cones to intersect worldline of observer

X first followed by Y and Z simultaneously

Only Y and Z light cones need to be seen.