Relativity Qs

1. This question is about length contraction and simultaneity.

(a) Define proper length.

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(b) A spaceship is travelling to the right at speed 0.75 c, through a tunnel which is open atboth ends. Observer A is standing at the centre of one side of the tunnel. Observer A, forwhom the tunnel is at rest, measures the length of the tunnel to be 240 m and the length ofthe spaceship to be 200 m. The diagram below shows this situation from the perspective ofobserver A.

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

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

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

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

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IB Questionbank Physics 1

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

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(c) Two sources of light are located at each end of the tunnel. The diagram below shows thissituation from the perspective of observer A.

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

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(Total 10 marks)


2. This question is about length contraction and simultaneity.

(a) Define proper length.

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(b) A spaceship is travelling to the right at speed 0.75 c, through a tunnel which is open atboth ends. Observer A is standing at the centre of one side of the tunnel. Observer A, forwhom the tunnel is at rest, measures the length of the tunnel to be 240 m and the length ofthe spaceship to be 200 m. The diagram below shows this situation from the perspective ofobserver A.

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

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

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

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

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

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(c) Two sources of light are located at each end of the tunnel. The diagram below shows thissituation from the perspective of observer A.

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

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(Total 10 marks)


3. This question is about relative velocities and energy at relativistic speeds.

Two identical rockets are moving along the same straight line as viewed from Earth.Rocket 1 is moving away from the Earth at speed 0.80 c relative to the Earth and rocket 2 ismoving away from rocket 1 at speed 0.60 c relative to rocket 1.

(a) Calculate the velocity of rocket 2 relative to the Earth, using the

(i) Galilean transformation equation.

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(ii) relativistic transformation equation.

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(b) Comment on your answers in (a).

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(c) The rest mass of rocket 1 is 1.0 × 103 kg. Determine the relativistic kinetic energy ofrocket 1, as measured by an observer on Earth.

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(Total 8 marks)

4. This question is about the Michelson–Morley experiment.

The diagram below shows the essential features of the apparatus used in the Michelson–Morleyexperiment.


(a) State and explain how this apparatus was used to try to measure the speed of the Earthrelative to the ether.

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(b) State the result of this experiment and explain how this result supports the special theoryof relativity.

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(Total 6 marks)


5. This question is about gravitational red-shift.

Two identical lasers are situated on the surface of the Earth. One is directed horizontally towards

observer A, who measures the frequency to be 4.62 × 1014 Hz. The other is directed vertically

upwards towards observer B, who is at a height of 1.00 × 102 m

(a) (i) State how the frequency as measured by observer B compares with the frequency asmeasured by observer A.

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(ii) Calculate the difference in frequency between the laser light as measured byobservers A and B.

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(iii) State one assumption that you made in (a)(ii).

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(b) The lasers are now placed on a spaceship, which is accelerating upwards at a constant rate

of 9.81 m s–2, far away from any other masses as shown below. The distance of observer

D from the laser is 1.00 × 102 m. Observer C is at the bottom of the spaceship.

Explain, with reference to the equivalence principle, the frequencies measured by observersC and D, as compared to observers A and B.

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(Total 6 marks)


6. This question is about relativity.

Carrie is in a spaceship that is travelling towards a star in a straight-line at constant velocity asobserved by Peter. Peter is at rest relative to the star.

(a) Carrie measures her spaceship to have a length of 100 m. Peter measures Carrie’sspaceship to have a length of 91 m.

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

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

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(b) 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.

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(c) 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.

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(Total 8 marks)

7. This question is about a light clock.

Describe the principle of a light clock.

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8. This question is about relativity.

Carrie is in a spaceship that is travelling towards a star in a straight-line at constant velocity asobserved by Peter. Peter is at rest relative to the star.

(a) Carrie measures her spaceship to have a length of 100 m. Peter measures Carrie’sspaceship to have a length of 91 m.

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

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

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(b) 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.

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(c) 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.

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(d) Carrie and Louise, two observers in a spaceship, view a light source placed close to Carrie.When the spaceship is travelling at a constant velocity, they both measure the frequency ofthe light source and obtain identical values.

The magnitude of the velocity of the spaceship increases.

State and explain any changes to the frequency of the light source, as measured by Louise,that occur during the acceleration.

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(Total 12 marks)


9. This question is about relativistic mechanics.

(a) Calculate the potential difference through which a proton, starting from rest, must beaccelerated for its mass–energy to be equal to three times its rest mass-energy.

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(b) Calculate the momentum of the proton after acceleration.

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(Total 6 marks)

10. This question is about the Michelson–Morley experiment.

(a) (i) Outline the purpose of the experiment.

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(ii) State and explain why Michelson and Morley rotated the apparatus through 90°.

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(b) State and explain the significance of the result of the experiment.

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(Total 6 marks)

11. This question is about spacetime.

(a) Describe what is meant by spacetime.

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(b) State the shape of the path in spacetime of a body

(i) moving at constant velocity.

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(ii) orbiting the Earth.

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(c) Explain how spacetime is used to describe the gravitational attraction between Earth and asatellite orbiting the Earth.

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(Total 6 marks)

12. This question is about relativistic kinematics.

(a) Define inertial frame of reference.

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(b) Ann and Sue are twins. Sue remains on Earth. Ann travels to the star Sirius in a spaceshipmoving at a speed of 0.80 c, as measured by Sue. The distance between Earth and Sirius is8.8 ly, as measured by Sue.

(i) Calculate the time elapsed, as measured by Sue, between Ann leaving Earth andreaching Sirius.

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(ii) State and explain whether Ann or Sue measures the proper time between Annleaving Earth and arriving at Sirius.

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(iii) Calculate the time taken for the spaceship to reach Sirius, as measured by Ann.

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(iv) As Ann approaches Sirius, she sends a radio message back to Sue. The distancebetween Sirius and Earth, as measured by Ann, is 5.28 ly. Determine the time, asmeasured by Ann, that it takes for the signal to reach Sue.

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(Total 10 marks)



(a) Define inertial frame of reference.

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(b) Ann and Sue are twins. Sue remains on Earth. Ann travels to the star Sirius in a spaceshipmoving at a speed of 0.80c, as measured by Sue. The distance between Earth and Sirius is8.8 ly, as measured by Sue.

(i) Calculate the time elapsed, as measured by Sue, between Ann leaving Earth andreaching Sirius.

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(ii) State and explain whether Ann or Sue measures the proper time between Annleaving Earth and arriving at Sirius.

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(iii) Calculate the time taken for the spaceship to reach Sirius, as measured by Ann.

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(iv) As Ann approaches Sirius, she sends a radio message back to Sue. The distancebetween Sirius and Earth, as measured by Ann, is 5.28 ly. Determine the time, asmeasured by Ann, that it takes for the signal to reach Sue.

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(c) As soon as Ann arrives at Sirius, she immediately turns around and returns to Earth at aspeed of 0.80c, as measured by Sue. The return journey gives rise to the twin paradox.

(i) State what is meant by the twin paradox.

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(ii) Outline how the twin paradox is resolved.

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(Total 13 marks)


14. This question is about relativistic mass and energy.

(a) Define rest mass.

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(b) Explain why particles cannot be accelerated to the speed c.

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(c) Two protons are travelling towards one another along the same straight line. The speed ofeach proton is 0.70c as measured by a laboratory observer. Calculate the relative velocityof approach, as measured in the frame of reference of one of the protons.

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(d) Determine the potential difference through which a proton must be accelerated in order toreach a speed of 0.70c.

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(Total 9 marks)

15. This question is about black holes.

(a) Define the Schwarzschild radius.

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(b) Calculate the Schwarzschild radius for an object of mass 2.0 × 1031 kg.

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(c) A starship is stationary just outside the event horizon of a black hole. A space station isalso stationary and is located far away from the black hole and any other massive object.

(i) The starship transmits a radio signal to the space station. Explain why the signalreceived at the space station is shifted to a lower frequency than the transmittedfrequency.

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(ii) The starship remains stationary just outside the event horizon for one hour asmeasured by an observer in the starship. The time elapsed, as measured by anobserver in the space station, is ten hours. Determine, in terms of the Schwarzschildradius Rs of the black hole, the distance of the starship from the event horizon of the

black hole.

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(Total 8 marks)



(a) State what is meant by proper length and proper time interval.

Proper length: ..............................................................................................................

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Proper time interval: ...................................................................................................

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(b) A spaceship moves with speed v = 0.600c relative to a space station. Two lasers, A and B,on the space station are 5.00 m apart as measured by the space station observers. Thegamma factor for a speed of v = 0.600c is γ =1.25.

(i) A radio signal is sent to the spaceship from the space station. The transmission lastsfor 6.00 s according to space station clocks. Calculate the duration of thetransmission according to the spaceship observers.

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(ii) Calculate the distance between the lasers A and B according to the spaceshipobservers.

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(c) The lasers in (b) are fired simultaneously according to the space station observers.Light from each laser makes a mark on the spaceship. The spaceship observers measurethe distance between the two marks to be 6.25 m. According to the spaceship observers

(i) state which laser was fired first.

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(ii) determine the difference in time between the firings of the two lasers.

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(Total 9 marks)

17. This question is about frames of reference and the speed of light.

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

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(b) The diagram shows two inertial frames of reference. Frame S′ moves with velocity v = 2.0

× 108 m s–1 relative to frame S along the x-x′-direction. When the origins of the two framescoincided, the clocks in both frames were set to show zero.

An event takes place at x = 5.0 × 103 m when the clocks in frame S show t = 3.0 × 10–5 s.

(i) By using Galilean transformation equations, determine the position and time of thisevent in the S′ frame of reference.

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(ii) A laser, at rest in the frame S′, emits a ray of light along the negative x′-direction(i.e. towards the left). The speed of the ray of light is measured to be c in the frameS′. Show, using the relativistic velocity addition formula, that the speed of the ray oflight in the frame S is also measured to be c.

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(c) Outline an experiment (other than the Michelson–Morley experiment) which providesevidence that the speed of light is independent of the speed of its source.

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(Total 8 marks)


(a) State what is meant by proper length and proper time interval.

Proper length: ..............................................................................................................

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Proper time interval: ....................................................................................................

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(b) A spaceship moves with speed v = 0.600c relative to a space station. Two lasers, A and B,on the space station are 5.00 m apart as measured by the space station observers. Thegamma factor for a speed of v = 0.600c is γ =1.25.


(i) A radio signal is sent to the spaceship from the space station. The transmission lastsfor 6.00 s according to space station clocks. Calculate the duration of thetransmission according to the spaceship observers.

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(ii) Calculate the distance between the lasers A and B according to the spaceshipobservers.

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(c) The lasers in (b) are fired simultaneously according to the space station observers.Light from each laser makes a mark on the spaceship. The spaceship observers measurethe distance between the two marks to be 6.25 m. According to the spaceship observers

(i) state which laser was fired first.

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(ii) determine the difference in time between the firings of the two lasers.

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(Total 9 marks)



The upsilon (Y0) is an unstable particle that decays into a tau (τ–) and an antitau (τ+) accordingto the reaction

Y0 → τ– + τ+

The Y0 is at rest relative to the laboratory when it decays. The momentum of the τ– relative to the

laboratory is 4.40 GeV c–1. The rest mass of the τ– and of the τ+ is 1.78 GeV c–2.

Determine the rest mass of the Y0.

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20. This question is about general relativity.

(a) State Einstein’s principle of equivalence.

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(b) A rocket is accelerating in the direction shown far from any masses.

A monochromatic beam of light is emitted from the floor F of the rocket. The beam isreceived at the ceiling C of the rocket. Suggest why the frequency of the light beammeasured at C is less than the frequency measured at F.

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(c) The diagram shows three identical rockets A, B and C.

Rocket A is accelerating in outer space with acceleration a.

Rocket B is at rest on the surface of a planet. The gravitational field strength at the surfaceof the planet is a.

Rocket C is falling freely above a planet. The gravitational field strength of the planet inthe region of the rocket is a.

Light is emitted from a source X on the left wall of each rocket. The receiver Y is directlyopposite X.

The initial direction of the light is parallel to the floor of the rocket.

Draw, for each rocket, the path of the light ray emitted from X according to an observer ineach rocket.

(3)


(d) Einstein’s theory of general relativity predicts that a massive body causes a curvature(bending) of space.

(i) Describe what is meant by the curvature of space.

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(ii) Describe how Eddington’s experiment provides evidence for the curvature of spacecaused by the Sun.

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(Total 10 marks)


21. This question is about a Galilean transformation and time dilation.

Ben is in a spaceship that is travelling in a straight-line with constant speed v as measured by Jillwho is in a space station.

Ben switches on a light pulse that bounces vertically (as observed by Ben) between twohorizontal mirrors M1 and M2 separated by a distance d. At the instant that the mirrors are

opposite Jill, the pulse is just leaving the mirror M2. The speed of light in air is c.

(a) On the diagram, sketch the path of the light pulse between M1 and M2 as observed by Jill.

(1)

(b) The time for the light pulse to travel from M2 to M1 as measured by Jill is Δt.

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

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(ii) Using a Galilean transformation, derive an expression for the length of the path ofthe light between M2 and M1.

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(c) State, according to special relativity, the length of the path of the light between M2 and M1

as measured by Jill in terms of c and ∆t.

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(d) The time for the pulse to travel from M2 to M1 as measured by Ben is ∆t′. Use your

answers to (b)(i) and (c) to derive a relationship between ∆t and ∆t′.

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

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(Total 10 marks)


22. This question is about a Galilean transformation and time dilation.

Ben is in a spaceship that is travelling in a straight-line with constant speed v as measured by Jillwho is in a space station.

Ben switches on a light pulse that bounces vertically (as observed by Ben) between twohorizontal mirrors M1 and M2 separated by a distance d. At the instant that the mirrors are

opposite Jill, the pulse is just leaving the mirror M2. The speed of light in air is c.

(a) On the diagram, sketch the path of the light pulse between M1 and M2 as observed by Jill.

(1)

(b) The time for the pulse to travel from M2 to M1 as measured by Jill is ∆t.

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

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(ii) Using a Galilean transformation, derive an expression for the length of the path ofthe light between M2 and M1.

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(c) State, according to special relativity, the length of the path of the light between M2 and M1

as measured by Jill in terms of c and ∆t.

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(d) The time for the pulse to travel from M2 to M1 as measured by Ben is ∆t′. Use your

answers to (b)(i) and (c) to derive a relationship between ∆t and ∆t′.

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

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(f) The length of a table in the spaceship is measured by Ben to be 1.8 m. Calculate the lengthof the table as measured by Jill.

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(g) The questions (e) and (f) introduce the concepts of time dilation and length contraction.Discuss how muon decay in the atmosphere provides experimental evidence for theseconcepts.

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(Total 16 marks)


23. This question is about pair production and relativistic mechanics.

A γ-photon of energy 2.46 MeV is travelling close to the nucleus of a gold atom. It converts into

an electron (e–)–positron (e+) pair. Each particle travels at 45° to the original direction of thephoton.

Immediately after the conversion, the kinetic energies of the electron and positron are equal. The

magnitude of the recoil momentum of the gold nucleus is 0.880 MeV c–1 and is in the direction ofthe photon.

(a) Calculate, immediately after the decay, the magnitude of the momentum of the electron.

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(b) Calculate the value V of the potential difference through which an electron at rest must beaccelerated in order to have the same magnitude of momentum as that in (a).

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(Total 6 marks)

24. This question is about general relativity.

Bob is standing on the floor of a spaceship and he throws a ball in a direction parallel to the floor.

Between leaving Bob’s hand and landing on the floor, the ball follows the path shown.

(a) State and explain whether, from the path followed by the ball, Bob can deduce that thespaceship is at rest on the surface of a planet.

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(b) Outline how the concept of spacetime can be used to explain the

(i) trajectory of the ball if the spaceship is at rest on the surface of a planet.

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(ii) nature of a black hole.

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(c) Calculate the radius that Earth would have to have in order for it to behave as a black hole.

The mass of Earth is 6.0 × 1024 kg.

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(Total 8 marks)


25. This question is about simultaneity.

Albert is at rest with respect to the ground. Hermann is in a carriage that is moving with speed vrelative to Albert in the direction shown. Two flashes of light are emitted from the back and thefront of the carriage. According to Hermann’s clock they arrive at Hermann’s positionsimultaneously.

Explain with reference to the concept of proper time, why the arrival of the light pulses atHermann will also be simultaneous to Albert.

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A spacecraft leaves Earth at a speed of 0.80c as measured by an observer on Earth. It headstowards, and continues beyond, a distant planet. The planet is 52 light years away from Earth asmeasured by an observer on Earth.

When the spacecraft leaves Earth, Amanda, one of the astronauts in the spacecraft, is 20 yearsold.

The Lorentz gamma factor for a speed of 0.80c is γ =

.3

5

(a) Calculate

(i) the time taken for the journey to the planet as measured by an observer on Earth.

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(ii) the distance between the Earth and the planet, as measured by Amanda.

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(iii) Amanda’s age as the spacecraft goes past the planet, according to Amanda.

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(b) As the spacecraft goes past the planet Amanda sends a radio signal to Earth.Calculate, as measured by the spacecraft observers, the time it takes for the signal to arriveat Earth.

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(Total 7 marks)

27. This question is about the Michelson-Morley experiment.

The diagram shows the type of arrangement that was used in the Michelson-Morley experiment.

The speed of light in free space is c and the orbital speed of Earth is v.


(a) State

(i) the purpose of the Michelson-Morley experiment.

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(ii) using a Galilean transformation, the speed of light along the path from the mirrorM1 to the semi-transparent mirror S in terms of c and v.

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(b) With the apparatus in the position shown, an interference pattern was observed in the fieldof view of the telescope. The apparatus was then rotated by 90º.

State

(i) what was expected to be observed.

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(ii) what was actually observed.

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(iii) how the result of the Michelson-Morley experiment is explained in the theory ofspecial relativity.

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(Total 5 marks)

28. This question is about simultaneity.

(a) Define what is meant by a proper time interval.

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(b) Albert is at rest with respect to the ground. Hermann is in a carriage that is moving withspeed v relative to Albert in the direction shown. Two flashes of light are emitted from theback and the front of the carriage. According to Hermann’s clock they arrive at Hermann’sposition simultaneously.

Explain with reference to the concept of proper time, why the arrival of the light pulses atHermann will also be simultaneous to Albert.

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(Total 4 marks)



A spacecraft leaves Earth at a speed of 0.80c as measured by an observer on Earth. It headstowards, and continues beyond, a distant planet. The planet is 52 light years away from Earth asmeasured by an observer on Earth.

When the spacecraft leaves Earth, Amanda, one of the astronauts in the spacecraft, is 20 yearsold.

The Lorentz gamma factor for a speed of 0.80c is γ =

.3

5

(a) Calculate

(i) the time taken for the journey to the planet as measured by an observer on Earth.

...........................................................................................................................(1)

(ii) the distance between the Earth and the planet, as measured by Amanda.

...........................................................................................................................

...........................................................................................................................(1)

(iii) Amanda’s age as the spacecraft goes past the planet, according to Amanda.

...........................................................................................................................

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(b) As the spacecraft goes past the planet Amanda sends a radio signal to Earth.Calculate, as measured by the spacecraft observers, the time it takes for the signal to arriveat Earth.

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(Total 7 marks)


The graph shows the variation with speed v of the total energy E of a particle called the kaon,

K–. The rest mass of the K– is 490 MeV c–2 and its electric charge is the same as that of theelectron.


(a) Suggest how the graph may be used in order to obtain the rest mass of the K–.

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(b) Use the graph to determine the potential difference required to accelerate a K– from rest toa speed of 0.95c.

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(c) The K– is unstable and decays into a muon and an antineutrino according to the reaction

K– →µ– + v

The rest mass of the muon is 105 MeV c–2. The rest mass of the antineutrino may beassumed to be zero.

The K– is at rest when it decays.

(i) Show that total energy is conserved if the momentum of the muon is 234 MeV c–1.

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(ii) Determine the kinetic energy of the muon.

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(Total 8 marks)

31. This question is about black holes

(a) Define what is meant by the Schwarzschild radius of a black hole.

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(b) A probe is at rest a small distance from a black hole of Schwarzschild radius 6.7 × 104 m.The probe emits monochromatic light pulses that are received by a spacecraft very farfrom the black hole.

According to a clock in the probe the pulses have a frequency of 4.5 × 1014 Hz, with onepulse being emitted every two seconds. The frequency of the light as measured by a clock

in the spacecraft is 3.0 × 1014 Hz.

(i) State the name of the phenomenon responsible for the observed change in thefrequency of the light.

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(ii) Determine, according to the spacecraft clock, how often the pulses are received bythe spacecraft.

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(iii) Determine the distance of the probe from the centre of the black hole.

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(Total 6 marks)

32. This question is about simultaneity and length measurement.

(a) One of the two postulates of special relativity states that “the laws of physics are the samefor all inertial observers”. State the other postulate.

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(b) Tom and Barbara are two observers each in a separate reference frame. The referenceframes are moving relative to each other in the same straight line with constant velocity.Two lamps L1 and L2 are operated by the same switch. Tom is at the mid-point between

the lamps as measured in his frame of reference.

The lamps and the switch are at rest relative to Tom.

Tom switches on the lamps and to him they light simultaneously. Explain, based on youranswer to (a), why the lamps will not light simultaneously, according to Barbara.

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(c) Tom measures the separation of L1 and L2 to be 1.5 m whereas Barbara measures the

separation to be 0.5 m.

(i) State and explain which observer measures the proper length between L1 and L2.

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(ii) Calculate, in terms of the free space speed of light c, the relative speed between Tomand Barbara.

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(iii) Sketch a graph to show how the relative speed v between Tom and Barbara varieswith the length L between L1 and L2 as measured by Barbara. The data point (1.5,0)

is shown. On the v axis, label the point v = c.

(3)

(Total 11 marks)

33. This question is about simultaneity and length measurement.

(a) One of the two postulates of special relativity states that “the laws of physics are the samefor all inertial observers”. State the other postulate.

......................................................................................................................................

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(b) Tom and Barbara are two observers each in a separate reference frame. The referenceframes are moving relative to each other in the same straight line with constant velocity.Two lamps L1 and L2 are operated by the same switch. Tom is at the mid-point between

the lamps as measured in his frame of reference.

The lamps and the switch are at rest relative to Tom.

Tom switches on the lamps and to him they light simultaneously. Explain, based on youranswer to (a), why the lamps will not light simultaneously, according to Barbara.

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(c) Tom measures the separation of L1 and L2 to be 1.5 m whereas Barbara measures the

separation to be 0.5 m.

(i) State and explain which observer measures the proper length between L1 and L2.

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(ii) Calculate, in terms of the free space speed of light c, the relative speed between Tomand Barbara.

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(iii) Sketch a graph to show how the relative speed v between Tom and Barbara varieswith the length L between L1 and L2 as measured by Barbara. The data point (1.5,

0) is shown. On the v axis, label the point v = c.

(3)


(d) In another situation, Tom is at rest on the surface of Earth. Barbara is in a spaceshiptravelling at constant velocity, with respect to Earth, towards a distant planetary system.After reaching the system she returns to Earth. Outline how this situation leads to theso-called twin paradox.

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(Total 14 marks)

34. This question is about some consequences of special relativity.

A proton is accelerated from rest through a potential difference V. After acceleration its speed isclose to that of the speed of light in free space c.

(a) On the axes sketch a graph to show the variation with speed v of the mass m of the proton.

(3)


(b) The potential difference V is 8.50 ×108 V. As measured in the laboratory frame ofreference,

(i) calculate the total energy of the proton after acceleration.

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(ii) show that the final speed of the proton is 0.852c.

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(c) After acceleration the proton approaches an antiproton (proton with a negative charge) thatis moving along the same straight line but in the opposite direction. The relative speed ofapproach of the particles, as measured in the laboratory frame of reference, is 0.987c.Deduce that the speed of the antiproton is about the same as the speed of the proton in(b)(ii).

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(Total 10 marks)


35. This question is about gravitational red-shift and black holes.

The concept of gravitational red-shift indicates that clocks run slower as they approach a blackhole.

(a) Describe what is meant by

(i) gravitational red-shift.

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(ii) spacetime.

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(iii) a black hole with reference to the concept of spacetime.

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(b) A particular black hole has a Schwarzschild radius R. A person at a distance of 2R fromthe event horizon of the black hole measures the time between two events to be 10 s.Deduce that for a person a very long way from the black hole the time between the eventswill be measured as 12 s.

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(Total 6 marks)

36. This question is about special relativity.

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

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(b) Aibhe is at rest in an inertial frame of reference A and Euan is at rest in an inertial frameof reference B. B is moving in the x-direction, with speed v, relative to A. The table T is atrest with respect to Aibhe.

Aibhe measures the length of T to be 1.5 m and Euan measures it to be 1.2 m.


(i) Explain which observer measures the proper length of T.

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(ii) Determine the speed v.

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(c) Two insects land at the same point on T. According to a clock at rest with respect toAibhe, one of the insects lands 2.4 seconds after the other. Calculate, according to Euan,the time interval between the landings of each insect.

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(d) Two other insects land at either end of the table. These two events may occur at the sametime as measured by one of the observers (Aibhe or Euan) but not to the other. Outline,with reference to the postulates of relativity, why these times differ.

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(Total 10 marks)

37. This question is about special relativity.


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(b) Aibhe is at rest in an inertial frame of reference A and Euan is at rest in an inertial frameof reference B. B is moving in the x-direction, with speed v, relative to A. The table T is atrest with respect to Aibhe.

Aibhe measures the length of T to be 1.5 m and Euan measures it to be 1.2 m.


(i) Explain which observer measures the proper length of T.

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(ii) Determine the speed v.

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(c) Two insects land at the same point on T. According to a clock at rest with respect toAibhe, one of the insects lands 2.4 seconds after the other. Calculate, according to Euan,the time interval between the landings of each insect.

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(d) Two other insects land at either end of the table. These two events may occur at the sametime as measured by one of the observers (Aibhe or Euan) but not to the other. Outline,with reference to the postulates of relativity, why these times differ.

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(e) Muons are produced in the upper atmosphere by the bombardment of cosmic rays. Themuons travel close to the speed of light and are unstable. They have a very short half-lifeas measured in a muon’s frame of reference. Explain how the detection of muons at thesurface of the Earth provides evidence for the special theory of relativity.

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(Total 13 marks)

38. This question is about relativistic energy and momentum.

(a) Particle A is at rest with respect to an observer. Another identical particle B is moving withrespect to the observer. Distinguish between the total energy of particle A and the totalenergy of particle B as measured by the observer.

Particle A: ....................................................................................................................

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Particle B: ....................................................................................................................

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(b) Two protons are travelling towards each other along the same straight line in a vacuum.

The speed of each proton, as measured in the laboratory frame of reference, is 0.960c.


(i) Calculate the relative speed of one proton with respect to the other proton.

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(ii) Show that the total energy of one of the protons, according to an observer at rest inthe laboratory, is 3.35 GeV.

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(c) The collision of the two protons results in the following reaction

p+ + p+ = p+ + n0 + π+

where π+ is a particle called a pion and has a rest mass of 140 MeV c–2. The total energyof the pion is 502 MeV. Determine, according to an observer at rest in the laboratory, the

(i) total energy of the proton formed plus the total energy of the neutron formed by thecollision.

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(ii) momentum of the pion.

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(d) The diagram shows the paths followed by the neutron and pion in (c).

The dotted line shows the path of the original collision of the protons in (b). On thediagram, draw the direction of the proton formed in the collision.

(1)(Total 11 marks)

39. This question is about spacetime and black holes.

(a) State what is meant by the concept of spacetime.

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(b) Outline, with reference to the motion of photons in spacetime, what is meant by theSchwarzschild radius.

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(c) The mass of the Sun is about 2 × 1030 kg. Show that, if the Sun were to become a blackhole, its radius would be about 3 km.

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(Total 6 marks)



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(b) A spacecraft is moving with a speed of 0.80c with respect to observers on Earth. After 6.0years of travel, according to the spacecraft clocks, the spacecraft arrives at a distant solarsystem.

(i) Calculate the time the journey has taken according to an observer on Earth.

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(ii) Calculate the distance between the Earth and the solar system according to anobserver on Earth.

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(iii) The spacecraft observers send a signal to Earth to announce that they have arrivedat the solar system. The spacecraft continues to move. Determine how long it willtake the signal to arrive on Earth according to the spacecraft observers.

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(Total 10 marks)



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(b) A spacecraft is moving with a speed of 0.80c with respect to observers on Earth. After 6.0years of travel, according to the spacecraft clocks, the spacecraft arrives at a distant solarsystem.

(i) Calculate the time the journey has taken according to an observer on Earth.

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(ii) Calculate the distance between the Earth and the solar system according to anobserver on Earth.

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(iii) The spacecraft observers send a signal to Earth to announce that they have arrivedat the solar system. The spacecraft continues to move. Determine how long it willtake the signal to arrive on Earth according to the spacecraft observers.

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(Total 10 marks)


42. Two electrons are travelling directly towards one another. Each has a speed of 0.80c relative to astationary observer. Calculate the relative velocity of approach, as measured in the frame ofreference of one of the electrons.

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43. This question is about mass-energy.

(a) Distinguish between the rest mass-energy of a particle and its total energy.

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(b) The rest mass of a proton is 938 MeV c–2. State the value of its rest mass-energy.

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(c) A proton is accelerated from rest through a potential difference V until it reaches a speed of0.980c. Determine the potential difference V as measured by an observer at rest in thelaboratory frame of reference.

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(Total 7 marks)

44. The diagram below illustrates the distortion of space by the gravitational field of a black hole.

(a) (i) Describe what is meant by the centre and the surface of a black hole.

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(ii) With reference to your answer in (a)(i), define the Schwarzschild radius.

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(iii) Calculate the Schwarzschild radius for an object having a mass of 2.0 ́ 1031 kg (tensolar masses).

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(iv) Science fiction frequently portrays black holes as objects that “swallow” everythingin the universe.

A spacecraft is travelling towards the object in (a)(iii) such that, if it continues in a

straight-line, its distance of closest approach would be about 107 m. By reference tothe diagram and your answer in (a)(iii), suggest whether the fate of the spacecraft isconsistent with the fate as portrayed in science fiction.

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(b) In 1979, Wahl, Carswell and Weymann discovered “two” very distant quasars separatedby a small angle. Spectroscopic examination of the images showed that they were identical.

Outline how these observations give support to the theory of General Relativity.

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(Total 10 marks)


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