SL Paper 3
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.
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.
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.
Identical twins, A and B, are initially on Earth. Twin A remains on Earth while twin B leaves the Earth at a speed of 0.6c for a return journey to a point three light years from Earth.
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.
An observer P sitting in a train moving at a speed v measures that his journey takes a time ΔtP. An observer Q at rest with respect to the ground measures that the journey takes a time ΔtQ.
According to Q there is an instant at which the train is completely within the tunnel.
At that instant two lights at the front and the back of the train are turned on simultaneously according to Q.
The spacetime diagram according to observer Q shows event B (back light turns on) and event F (front light turns on).
State which of the two time intervals is a proper time.
Calculate the speed v of the train for the ratio .
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 and space 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.
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.
A rocket of proper length 450 m is approaching a space station whose proper length is 9.0 km. The speed of the rocket relative to the space station is 0.80c.
X is an observer at rest in the space station.
Two lamps at opposite ends of the space station turn on at the same time according to X. Using a Lorentz transformation, determine, according to an observer at rest in the rocket,
The rocket carries a different lamp. Event 1 is the flash of the rocket’s lamp occurring at the origin of both reference frames. Event 2 is the flash of the rocket’s lamp at time ct' = 1.0 m according to the rocket. The coordinates for event 2 for observers in the space station are x and ct.
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.
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 = 5.0. Discuss, with suitable calculations, how this experiment provides evidence for time dilation.
A spaceship S leaves the Earth with a speed v = 0.80c. The spacetime diagram for the Earth is shown. A clock on the Earth and a clock on the spaceship are synchronized at the origin of the spacetime diagram.
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.
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.
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.
Two parallel current-carrying wires have equal currents in the same direction. There is an attractive force between the wires.
Maxwell’s equations led to the constancy of the speed of light. Identify what Maxwell’s equations describe.
State a postulate that is the same for both special relativity and Galilean relativity.
Identify the nature of the attractive force recorded by an observer stationary with respect to the wires.
A second observer moves at the drift velocity of the electron current in the wires. Discuss how this observer accounts for the force between the wires.
A spaceship is travelling at , away from Earth. It launches a probe away from Earth, at relative to the spaceship. An observer on the probe measures the length of the probe to be .
The Lorentz transformations assume that the speed of light is constant. Outline what the Galilean transformations assume.
Deduce the length of the probe as measured by an observer in the spaceship.
Explain which of the lengths is the proper length.
Calculate the speed of the probe in terms of , relative to Earth.
Two rockets, A and B, are moving towards each other on the same path. From the frame of reference of the Earth, an observer measures the speed of A to be 0.6c and the speed of B to be 0.4c. According to the observer on Earth, the distance between A and B is 6.0 x 108 m.
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′ =
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.
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 0.995c with respect to the Earth. When measured at rest the average lifetime of the muons is 2.1 μs.
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.
An observer on Earth watches rocket A travel away from Earth at a speed of 0.80c. The spacetime diagram shows the worldline of rocket A in the frame of reference of the Earth observer who is at rest at x = 0.
Another rocket, B, departs from the same location as A but later than A at ct = 1.2 km according to the Earth observer. Rocket B travels at a constant speed of 0.60c in the opposite direction to A according to the Earth observer.
Rocket A and rocket B both emit a flash of light that are received simultaneously by the Earth observer. Rocket A emits the flash of light at a time coordinate ct = 1.8 km according to the Earth observer.
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.
Observer A detects the creation (event 1) and decay (event 2) of a nuclear particle. After creation, the particle moves at a constant speed relative to A. As measured by A, the distance between the events is 15 m and the time between the events is 9.0 × 10–8 s.
Observer B moves with the particle.
For event 1 and event 2,
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.
A train is passing through a tunnel of proper length 80 m. The proper length of the train is 100 m. According to an observer at rest relative to the tunnel, when the front of the train coincides with one end of the tunnel, the rear of the train coincides with the other end of the tunnel.
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 diagram shows the axes for two inertial reference frames. Frame S represents the ground and frame S′ is a box that moves to the right relative to S with speed v.
When the origins of the two frames coincide all clocks show zero. At that instant a beam of light of speed c is emitted from the left wall of the box towards the right wall. The box has proper length L. Consider the event E = light arrives at the right wall of the box.
Using Galilean relativity,
State what is meant by a reference frame.
explain why the time coordinate of E in frame S is .
hence show that the space coordinate of E in frame S is .
A rocket of proper length 120 m moves to the right with speed 0.82c relative to the ground.
A probe is released from the back of the rocket at speed 0.40c relative to the rocket.
Calculate the speed of the probe relative to the ground.
Determine the time it takes the probe to reach the front of the rocket according to an observer at rest in the rocket.
Determine the time it takes the probe to reach the front of the rocket according to an observer at rest on the ground.
The spacetime diagram shows the axes of an inertial reference frame S and the axes of a second inertial reference frame S′ that moves relative to S with speed 0.745c. When clocks in both frames show zero the origins of the two frames coincide.
Event E has coordinates x = 1 m and ct = 0 in frame S. Show that in frame S′ the space coordinate and time coordinate of event E are
A rod at rest in frame S has proper length 1.0 m. At t = 0 the left-hand end of the rod is at x = 0 and the right-hand end is at x = 1.0 m.
x′ = 1.5 m.
ct′ = –1.1 m.
Label, on the diagram, the space coordinate of event E in the S′ frame. Label this event with the letter P.
Label, on the diagram, the event that has coordinates x′ = 1.0 m and ct′ = 0. Label this event with the letter Q.
Using the spacetime diagram, outline without calculation, why observers in frame S′ measure the length of the
rod to be less than 1.0 m.
Using the spacetime diagram, estimate, in m, the length of this rod in the S′ frame.
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.
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.
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 0.75c and the speed of rocket B is 0.50c.
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.
When a spaceship passes the Earth, an observer on the Earth and an observer on the spaceship both start clocks. The initial time on both clocks is 12 midnight. The spaceship is travelling at a constant velocity with γ = 1.25. A space station is stationary relative to the Earth and carries clocks that also read Earth time.
Some of the radio signal is reflected at the surface of the Earth and this reflected signal is later detected at the spaceship. The detection of this signal is event B. The spacetime diagram is shown for the Earth, showing the space station and the spaceship. Both axes are drawn to the same scale.
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.
The speed of a spaceship is measured to be 0.50c by an observer at rest in the Earth’s reference frame.
Define an inertial reference frame.
As the spaceship passes the Earth it emits a flash of light that travels in the same direction as the spaceship with speed c as measured by an observer on the spaceship. Calculate, according to the Galilean transformation, the speed of the light in the Earth’s reference frame.
Use your answer to (a)(ii) to describe the paradigm shift that Einstein’s theory of special relativity produced.
A spaceship moves away from the Earth in the direction of a nearby planet. An observer on the Earth determines the planet is 4 ly from the Earth. The spacetime diagram for the Earth’s reference frame shows the worldline of the spaceship. Assume the clock on the Earth, the clock on the planet, and the clock on the spaceship were all synchronized when ct = 0.
Show, using the spacetime diagram, that the speed of the spaceship relative to the Earth is 0.80c.
Label, with the letter E, the event of the spaceship going past the planet.
Determine, according to an observer on the spaceship as the spaceship passes the planet, the time shown by the clock on the spaceship.
Determine, according to an observer on the spaceship as the spaceship passes the planet, the time shown by the clock on the planet.
On passing the planet a probe containing the spaceship’s clock and an astronaut is sent back to Earth at a speed of 0.80c relative to Earth. Suggest, for this situation, how the twin paradox arises and how it is resolved.
A train of proper length 85 m moves with speed 0.60c relative to a stationary observer on a platform.
Define proper length.
In the reference frame of the train a ball travels with speed 0.50c from the back to the front of the train, as the train passes the platform. Calculate the time taken for the ball to reach the front of the train in the reference frame of the train.
In the reference frame of the train a ball travels with speed 0.50c from the back to the front of the train, as the train passes the platform. Calculate the time taken for the ball to reach the front of the train in the reference frame of the platform.
A rocket moving with speed v relative to the ground emits a flash of light in the backward direction.
An observer in the rocket measures the speed of the flash of light to be c.
State the speed of the flash of light according to an observer on the ground using Galilean relativity.
State the speed of the flash of light according to an observer on the ground using Maxwell’s theory of electromagnetism.
State the speed of the flash of light according to an observer on the ground using Einstein’s theory of relativity.
The diagram shows space and time axes and ct for an observer at rest with respect to a galaxy. A spacecraft moving through the galaxy has space and time axes ′ and ct′.
A rocket is launched towards the right from the spacecraft when it is at the origin of the axes. This is labelled event 1 on the spacetime diagram. Event 2 is an asteroid exploding at = 100 ly and ct = 20 ly.
Plot, on the axes, the point corresponding to event 2.
Suggest whether the rocket launched by the spacecraft might be the cause of the explosion of the asteroid.
Show that the value of the invariant spacetime interval between events 1 and 2 is 9600 ly2.
An observer in the spacecraft measures that events 1 and 2 are a distance of 120 ly apart. Determine, according to the spacecraft observer, the time between events 1 and 2.
Using the spacetime diagram, determine which event occurred first for the spacecraft observer, event 1 or event 2.
Determine, using the diagram, the speed of the spacecraft relative to the galaxy.
A long straight current-carrying wire is at rest in a laboratory. A negatively-charged particle P outside the wire moves parallel to the current with constant velocity v relative to the laboratory.
In the reference frame of the laboratory the particle P experiences a repulsive force away from the wire.
One of the two postulates of special relativity states that the speed of light in a vacuum is the same for all observers in inertial reference frames. State the other postulate of special relativity.
State the nature of the force on the particle P in the reference frame of the laboratory.
Deduce, using your answer to part (a), the nature of the force that acts on the particle P in the rest frame of P.
Explain how the force in part (b)(ii) arises.
The velocity of P is 0.30c relative to the laboratory. A second particle Q moves at a velocity of 0.80c relative to the laboratory.
Calculate the speed of Q relative to P.
Muons are created at a height of 3230 m above the Earth’s surface. The muons move vertically downward at a speed of 0.980 c relative to the Earth’s surface. The gamma factor for this speed is 5.00. The half-life of a muon in its rest frame is 2.20 μs.
Estimate in the Earth frame the fraction of the original muons that will reach the Earth’s surface before decaying according to Newtonian mechanics.
Estimate in the Earth frame the fraction of the original muons that will reach the Earth’s surface before decaying according to special relativity.
Demonstrate how an observer moving with the same velocity as the muons accounts for the answer to (a)(ii).
The spacetime diagram is in the reference frame of an observer O on Earth. Observer O and spaceship A are at the origin of the spacetime diagram when time . The worldline for spaceship A is shown.
Event E is the emission of a flash of light. Observer O sees light from the flash when years and calculates that event E is away, in the positive direction.
Calculate in terms of the velocity of spaceship A relative to observer O.
Draw the axis for the reference frame of spaceship A.
Plot the event E on the spacetime diagram and label it E.
Determine the time, according to spaceship A, when light from event E was observed on spaceship A.
A train is moving across a bridge with a speed v = 0.40c. Observer A is at rest in the train. Observer B is at rest with respect to the bridge.
The length of the bridge LB according to observer B is 2.0 km.
According to observer B, two lamps at opposite ends of the bridge are turned on simultaneously as observer A crosses the bridge. Event X is the lamp at one end of the bridge turning on. Event Y is the lamp at the other end of the bridge turning on.
Events X and Y are shown on the spacetime diagram. The space and time axes of the reference frame for observer B are and ct. The line labelled ct' is the worldline of observer A.
Calculate, for observer A, the length LA of the bridge
Calculate, for observer A, the time taken to cross the bridge.
Outline why LB is the proper length of the bridge.
Draw, on the spacetime diagram, the space axis for the reference frame of observer A. Label this axis '.
Demonstrate using the diagram which lamp, according to observer A, was turned on first.
Demonstrate, using the diagram, which lamp observer A observes to light first.
Determine the time, according to observer A, between X and Y.
Two protons are moving to the right with the same speed v with respect to an observer at rest in the laboratory frame.
Outline why there is an attractive magnetic force on each proton in the laboratory frame.
Explain why there is no magnetic force on each proton in its own rest frame.
Explain why there must be a resultant repulsive force on the protons in all reference frames.
Outline the conclusion from Maxwell’s work on electromagnetism that led to one of the postulates of special relativity.