Rocket A and rocket B are travelling in opposite directions from the Earth along the same straight line.

In the reference frame of the Earth, the speed of rocket A is 0.75*c *and the speed of rocket B is 0.50*c*.

a.i.

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.

**Answer/Explanation**

## Markscheme

a.i.

1.25*c*

**[1 mark]**

*ALTERNATIVE 1*

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

0.91*c*

*ALTERNATIVE 2*

\(u’ = \frac{{ – 0.50 – 0.75}}{{1 – ( – 0.5 \times 0.75)}}c\)

–0.91*c*

*[2 marks]*

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

*OWTTE*

*[1 mark]*

Muons are created in the upper atmosphere of the Earth at an altitude of 10 km above the surface. The muons travel vertically down at a speed of 0.995*c *with respect to the Earth. When measured at rest the average lifetime of the muons is 2.1 μs.

a.i.

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.

**Answer/Explanation**

## Markscheme

a.i.

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

*Do not accept 10*^{4}*/c = 33 μ**s**.*

*[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]**

The diagram shows the motion of the electrons in a metal wire carrying an electric current as seen by an observer X at rest with respect to the wire. The distance between adjacent positive charges is *d*.

Observer Y is at rest with respect to the electrons.

a.

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.

**Answer/Explanation**

## Markscheme

a.

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]*

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

**Answer/Explanation**

## Markscheme

light is an EM wave

speed of light is independent of the source/observer

a.

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.

**Answer/Explanation**

## Markscheme

a.

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]*

A long current-carrying wire is at rest in the reference frame S of the laboratory. A positively charged particle P outside the wire moves with velocity *v* relative to S. The electrons making up the current in the wire move with the same velocity* v* relative to S.

a.

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.

**Answer/Explanation**

## Markscheme

a.

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]**

An electron X is moving parallel to a current-carrying wire. The positive ions and the wire are fixed in the reference frame of the laboratory. The drift speed of the free electrons in the wire is the same as the speed of the external electron X.

a.

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.

**Answer/Explanation**

## Markscheme

a.

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 biithe 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*

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.

**Answer/Explanation**

## Markscheme

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

*relativistic/Lorentz effects also present*

**OR**One of the postulates of special relativity states that the laws of physics are the same in all inertial frames of reference.

a.

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.

**Answer/Explanation**

## Markscheme

a.

not being accelerated**OR**

not subject to an unbalanced force**OR**

where Newton’s laws apply

(i) *c*

(ii) *c*+*v*

This question is about relativistic kinematics.

A spacecraft is flying in a straight line above a base station at a speed of 0.8*c*.

Suzanne is inside the spacecraft and Juan is on the base station.

While moving away from the base station, Suzanne observes another spacecraft travelling towards her at a speed of 0.8*c*. 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 0.976*c*. 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.

**Answer/Explanation**

## Markscheme

1.6*c*;

(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.*

## Examiners report

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.

This question is about relativistic kinematics.

Speedy is in a spacecraft being chased by Police Officer Sylvester. In Officer Sylvester’s frame of reference, Speedy is moving directly towards Officer Sylvester at 0.25c.

a.

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^{–8}s.

(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.

**Answer/Explanation**

## Markscheme

a.

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) \(y = \left( {\frac{1}{{\sqrt {1 – \frac{{{v^2}}}{{{c^2}}}} }} = } \right)1.15\);

Δ*t*=1.15 x Δ*t*_{0};

1.48 x 10^{-8}(s);

## Examiners report

a.

There were some good, clear answers to (a) but there were many vague statements about “point of view”.

There were also some good answers to (b) but most candidates struggled. It was rarely stated that light travels at the same speed for all observers.

(i) was well done and

(ii) was very well done.

This question is about time dilation.

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 1.80×10^{11}m.

a.

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.

**Answer/Explanation**

## Markscheme

a.

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

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;