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North Berwick High School

Higher Department of Physics

Unit 1 Our Dynamic Universe

Section 5

Section 5 Special Relativity

Note Making

Make a dictionary with the meanings of any new words.

Special Relativity 1. What is meant by a ? 2. Write a short note on the leading up to Einstein's theories (just include the main dates).

The principles

1. State Einstein's starting principles.

Time dilation

1. State what is meant by dilation. 2. Copy the formula and take note of the . 3. Copy the example. 4. Explain why we do not notice these events in everyday life. 5. Explain how are able to be detected at the surface of the Earth.

Length contraction (also known as Lorentz contraction)

1. Copy the formula.

Section 5 Special Relativity

Contents

Content Statements...... 1 Special Relativity...... 2 2 + 2 = 3...... 2 History...... 2 The principles...... 4 Time Dilation...... 4 Why do we not notice these time differences in everyday life?...... 7 Length contraction...... 8 ...... 9 Section 5: Special relativity Problems...... 10

Content Statements

Contents Notes Context

Introduction to Relativity introduced through Newtonian Relativity can be special relativity. , experienced in an intuitive Newtonian Relativity and the way. Examples include concept of absolute . walking in a moving train and moving sound sources. Experimental and theoretical considerations (details not At high speeds, non-intuitive required) leads to the relativistic effects are conclusion that the speed of observed. Length contraction light is the same for all and time dilation can be observers. studied using suitable animations. The constancy of the speed of Experimental verification light led Einstein to postulate includes detection at that space and time for a the surface of the Earth and moving object are changed accurate time measurements relative to a stationary on airborne . observer.

Length contraction and time The time dilation equation dilation. can be derived from the geometrical consideration of a light beam moving relative to a stationary observer.

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Section 5 Special Relativity

Einstein originally proposed his theory of special relativity in 1905 and it is often taken as the beginning of modern physics. It was a watershed in the development of physics because, starting from two assumptions, it made possible simpler and more consistent explanations of physics up to that time. It also opened the door to whole new ways of thinking about the Universe. This first theory of Einstein is termed ‘special’ because it only considers frames of reference moving at constant speed. His later general considers what happens if they are accelerating.

2 + 2 = 3

Imagine you are on a train going at 60 mph: You are travelling at 60 mph when viewed by an observer at the side of the tracks, but you are at rest when viewed by the passenger sitting opposite you. These two points of view are known as frames of reference in physics, and relativity is all about how the physics measured in these frames of reference compares. If the observer was on a train travelling at 60 mph in the opposite direction then in their frame of reference you would be travelling at 120 mph. This works well for objects travelling at low speed but we find that it does not work so well for objects moving at speeds close to the . For instance, if two observers were passing each other in opposite directions at 2.0 × 10 8 ms–1 (from the reference frame of a stationary bystander) they would see the other travelling at 2.8 × 10 8 ms–1 rather than 4.0 × 108 ms–1, as you might expect.

History

The idea of relativity has been around since 1632, when Galileo was trying to argue in favour of the rotation of the Earth around the sun. He considered the question of whether a scientist performing experiments in a sealed room on board a ship would notice any difference in the results if the ship was travelling at constant speed or at rest. Nowadays we might consider a train or aeroplane as the moving reference frame but it is worthwhile to note that the measurements we make in our laboratories are not in a universally stationary reference frame but on a rotating Earth (~950 km/h in Scotland) orbiting the Sun (~112,000 km/h), which is itself orbiting the centre of the galaxy (~800,000 km/h).

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When Newton first proposed his laws of motion in 1687, he adhered to Galileo's . However, this idea came under question in the 19th , particularly when explaining how light travels. In 1864, James Clerk - Maxwell developed a theory of electromagnetism, which stated that light was a wave that travelled through some yet to be discovered universal medium called the aether (pronounced Eether). This aether could be considered a stationary master frame of reference that spread throughout the whole universe, including the vacuum of space.

In 1881, JJ Thomson (who also discovered the ) noticed that the of an object in motion increased. His work and Maxwell’s were developed further to show that the increase in mass depended on the speed of motion. Heaviside and Searle (1888–1897) also observed that a moving electromagnetic field contracted in the direction of travel. These experimental observations and others could be consistent with the widely accepted theory that included the idea of the aether. However, the aether itself was proving difficult to find and there was also much debate about the nature of the aether itself. Was it stationary or was it affected by the motion of objects through it, a bit like a boat dragging some water along with it?

A very important experiment was performed by Michelson and Morley in 1887, where light was sent in two beams at right angles to each other to mirrors, so that they could recombine again at the centre. Because of the of the Earth around the Sun and its movement around the centre of the galaxy we would expect to see differences in the measurements as the Earth changed direction through the aether. However, this is now one of the most famous ‘failed’ experiments of physics as the two men were not able to prove the existence of the aether. Rather than accept that the aether didn’t actually exist, the scientists of their considered that this was either down to lack of accuracy in the apparatus or that there were other theoretical reasons why the experiment didn’t work.

However, there were a small number of scientists such as Lorentz and Poincaré who weren’t wholly convinced by the aether concept and were developing alternative theories. Einstein was also aware of Planck’s work in quantum physics, which stated that light could be thought of as a particle rather than a wave, an idea which did away with the need for an aether. This was also such a massive jump in thinking that there was a feeling that all the pillars of classical physics were open to challenge. So Einstein did not come up with his ideas in isolation but was widely influenced by other thinkers of his time. He was not an established scientist at this time but his work as a patent clerk involved a lot of study into synchronising clocks throughout the world, so he was often dealing with the relative nature of measurements of time. Although (or perhaps because) he had no access to scientific equipment he would consider the logical consequences of scientific assumptions. He called these his ‘thought experiments’ or Gedanken in German. It is perhaps this unique combination of

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intellect and imagination that produced the most famous scientist of the 20th century.

The principles

1. When two observers are moving at constant speeds relative to one another, they will observe the same laws of physics.

2. The speed of light (in a vacuum) is the same for all observers. This means that no matter how fast you go, you can never catch up with a beam of light, since it always travels at 3.0 × 10 8 m s–1 relative to you. If you (or anything made of matter) were able to travel as fast as light, then light couldn’t be moving away from you at that speed and that wouldn’t make sense – we can never see stationary light.

Einstein started with just these two principles. These were two almost philosophical statements, which he could not prove to be true, but using them he was able to derive his theory of special relativity. The consequences of this theory could then be tested against what we observe to be true in the universe. It was also impressive that Einstein was able to derive all the results of his predecessors using just a few pages of working rather than the volumes of scientific writing they had needed.

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Time dilation

A very simple thought experiment shows that one consequence of the speed of light being the same for all observers is that time experienced by all observers is not necessarily the same. There is no universal that we can all refer to – we can simply make measurements of time as we experience it.

Imagine you are on a spaceship travelling at constant speed (v) relative to a colleague on a space station. You are investigating a timing device based on the fact that the speed of light is constant. You fire a beam of light at a sensor and time how long it takes to arrive. From your point of view the light has less (l) to travel than from the point of view of your colleague on the space station (d). If you both observe the speed of light as the same then you cannot agree on when it arrives, ie both of you experience time in a different way.

movement of sensor sensor spaceship

l d

light light source source

Your point of view, Your stationary where the source colleague’s point of and sensor are view, where the moving along sensor moves after together the light is fired

The time for the experiment as observed by your stationary colleague (t') is greater than the time observed by you when moving with the sensor (t), ie what you might observe as taking 1 could appear to take 2 to your stationary colleague. Note that you would be unaware of any difference until you were able to meet up with your colleague again. Your body processes would continue at the same rate.

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The formula linking these can be shown to be:

t t' v 2 1 c 2

Note this is often written as

t' t

where is known as the Lorentz Factor. It is used often in the study of special relativity and is given by:

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v 2 1 c 2

Worked example

A rocket is travelling at a constant 2.7 × 10 8 m s–1 compared to an observer on Earth. The pilot measures the journey as taking 240 . How long did the journey take when measured from Earth?

Solution: t = 240 minutes, v = 2.7 × 108 m s–1, c = 3 × 108 m s–1, t' = ?

t t' v 2 1 c 2

240 t' (2 7 108 ) 2 1 (3 108 ) 2 t' = 550 minutes

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An observer on Earth would measure the journey as taking 550 minutes, ie 550 minutes would have passed from their point of reference.

v2 v 2 Note: is the same as , which can make the calculation simpler, c2 c particularly if the speed is given in the form of a fraction of the speed of light, eg the speed in the above example could have been given as 0.9 or 9/10 of the speed of light. The calculation can then be written as:

240 240 t ' 550minutes 1 0 92 1 0 81

Why do we not notice these time differences in everyday life?

A graph of the Lorentz factor versus speed (measured as a multiple of the speed of light) is shown below.

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We can see that for small speeds (ie less than 0.1 times the speed of light) the Lorentz factor is approximately 1 and relativistic effects are negligibly small. Even 0.1 times the speed of light is 300,000 m s –1 or 1,080,000 km h–1 or about 675,000 mph – a tremendously fast speed compared to everyday life.

However, the speed of satellites is fast enough that even these small changes will add up over time and affect the synchronisation of global positioning systems (GPS) and television broadcasters with users on the Earth. They have to be specially programmed to adapt for the effects of special relativity (and also , which is not covered here). Very precise measurements of these small changes in time have been performed on fast-flying aircraft and agree with predicted results within experimental error.

Further evidence in support of special relativity comes from the field of Particle Physics, in the form of the detection of a particle called a muon at the surface of the Earth. (Hopefully you will be able to build your own detector for these during the Particle Physics topic.) These are produced in the upper layers of the atmosphere by cosmic rays (high- protons from space). Their measured lifetime is about 2·2 μs and their speed is 99·9653% of the speed of light. Using speed, distance and time calculations (Classical Physics)means they should travel about 600 m in their lifetime. However, muons are produced around 25 km above the surface of the Earth so they should not be detected at the surface. The fact that they are can be explained using time dilation. In the frame of reference of the muons, they survive for about 2·2 μs, but from our perspective this time is 84 μs because of to time dilation (you can check the calculations for yourself). This gives muons enough time to travel to the surface of the Earth.

Length contraction

Another implication of Einstein’s theory is the shortening of length when an object is moving. Consider the muons discussed above. Their large speed means they experience a longer lifetime due to time dilation. An equivalent way of thinking about this is that the muons experience the height of the atmosphere as smaller (or contracted) by the same ratio as the time has increased (or dilated). A symmetrical formula for time dilation can be derived. Note that the contraction only takes place in the direction that the object is travelling:

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v 2 l' l 1 c 2

where l is the length as measured by the moving observer and l' is that measured by the stationary observer.

Ladder paradox

There is an apparent paradox thrown up by special relativity: consider a ladder that is just longer than a garage. If we fly the ladder at high speed through the garage does length contraction mean that from our stationary perspective it fits inside the garage? How can this be reconciled with the fact that from the ladder’s reference frame the garage appears even shorter as it moves towards the ladder? The key to this question is simultaneity, i.e. whether different reference frames can agree on the time of particular events. In order for the ladder to fit in the garage the front of the ladder must be inside at the same time as the back of the ladder. Because of time dilation, the stationary observer (you) and a moving observer on the ladder cannot agree on when the front of the ladder reaches the far end of the garage or the rear of the ladder reaches the front. If you work out the equations carefully then you can show that even when the ladder is contracted, the front of the ladder and the back of the ladder will not both be inside the garage at the same time!

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Section 5: Special relativity Problems

Relativity – Fundamental principles

1. A river flows at a constant speed of 0·5 m s 1 south. A canoeist is able to row at a constant speed of 1·5 m s 1.

(a) Determine the of the canoeist relative to the river bank when the canoeist is moving upstream. (b) Determine the velocity of the canoeist relative to the river bank when the canoeist is moving downstream.

2. In an airport, passengers use a moving walkway. The moving walkway is travelling at a constant speed of 0·8 m s 1 and is travelling east. For the following people, determine the velocity of the person relative to the ground:

(a) a woman standing at rest on the walkway (b) a man walking at 2·0 m s 1 in the same direction as the walkway is moving (c) a boy running west at 3·0 m s 1.

3. The steps of an escalator move at a steady speed of 1·0 m s 1 relative to the stationary side of the escalator.

(a) A man walks up the steps of the escalator at 2·0 m s 1. Determine the speed of the man relative to the side of the escalator. (b) A boy runs down the steps of the escalator at 3·0 m s 1. Determine the speed of the boy relative to the side of the escalator.

4. In the following sentences the words represented by the letters A, B, C, D, E, F and G are missing:

In _____A____ Theory of Special Relativity the laws of physics are the _____B____ for all observers, at rest or moving at constant velocity with respect to each other ie _____C____ acceleration.

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An observer, at rest or moving at constant _____D____ has their own frame of reference. In all frames of reference the _____E____, c, remains the same regardless of whether the source or observer is in motion. Einstein’s principles that the laws of physics and the speed of light are the same for all observers leads to the conclusion that moving clocks run _____F____ (time dilation) and moving objects are _____G____ (length contraction).

Match each letter with the correct word from the list below:

acceleration different Einstein’s fast lengthened Newton’s same shortened slow speed of light velocity zero

5. An observer at rest on the Earth sees an aeroplane fly overhead at a constant speed of 2000 km h 1. At what speed, in km h 1, does the pilot of the aeroplane see the Earth moving?

6. A scientist is in a windowless lift. Can the scientist determine whether the lift is moving with a:

(a) velocity (b) uniform acceleration?

7. Spaceship A is moving at a speed of 2·4 × 108 m s 1. It sends out a light beam in the forwards direction. Meanwhile another spaceship B is moving towards spaceship A at a speed of 2·4 × 108 m s 1. At what speed does spaceship B see the light beam from spaceship A pass?

8. A spacecraft is travelling at a constant speed of 7·5 × 107 m s 1. It emits a pulse of light when it is 3·0 × 1010 m from the Earth as measured by an observer on the Earth. Calculate the time taken for the pulse of light to reach the Earth according to a clock on the Earth when the spacecraft is moving:

(a) away from the Earth (b) towards the Earth.

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9. A spaceship is travelling away from the Earth at a constant speed of 1·5 × 108 m s 1. A light pulse is emitted by a lamp on the Earth and travels towards the spaceship. Find the speed of the light pulse according to an observer on:

(a) the Earth (b) the spaceship.

10. Convert the following fraction of the speed of light into a value in m s 1:

(a) 0·1 c (b) 0·5 c (c) 0·6 c (d) 0·8 c

11. Convert the following speeds into a fraction of the speed of light:

(a) 3·0 × 108 m s 1 (b) 2·0 × 108 m s 1 (c) 1·5 × 108 m s 1 (d) 1·0 × 108 m s 1

Relativity – Time dilation

1. Write down the relationship involving the t and dilated time t’ between two events which are observed in two different frames of reference moving at a speed, v, relative to one another (where the proper time is the time measured by an observer at rest with respect to the two events and the dilated time is the time measured by another observer moving at a speed, v, relative to the two events).

2. In the table shown, use the relativity equation for time dilation to calculate the value of each missing quantity (a) to (f) for an object moving at a constant speed relative to the Earth.

Dilated time Proper time Speed of object / m s 1 (a) 20 h 1·00 × 108 (b) 10 2·25 × 108 1400 s (c) 2·00 × 108 1.40 × 10 4 s (d) 1·00 × 108 84 s 60 s (e) 21 minutes 20 minutes (f)

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3. Two observers P and Q synchronise their watches at 11.00 am just as observer Q passes the Earth at a speed of 2 × 108 m s 1.

(a) At 11.15 am according to observer P’s watch, observer P looks at Q’s watch through a telescope. Calculate the time, to the nearest , that observer P sees on Q’s watch. (b) At 11.15 am according to observer Q’s watch, observer Q looks at P’s watch through a telescope. Calculate the time, to the nearest minute, that observer Q sees on P’s watch.

4. The lifetime of a star is 10 billion as measured by an observer at rest with respect to the star. The star is moving away from the Earth at a speed of 0·81 c. Calculate the lifetime of the star according to an observer on the Earth.

5. A spacecraft moving with a constant speed of 0·75 c passes the Earth. An on the spacecraft measures the time taken for Usain Bolt to run 100 m in the sprint final at the 2008 Olympic Games. The astronaut measures this time to be 14·65 s. Calculate Usain Bolt’s winning time as measured on the Earth.

6. A scientist in the laboratory measures the time taken for a nuclear reaction to occur in an atom. When the atom is travelling at 8·0 × 107 m s 1 the reaction takes 4·0 × 10 4 s. Calculate the time for the reaction to occur when the atom is at rest.

7. The light beam from a lighthouse sweeps its beam of light around in a circle once every 10 s. To an astronaut on a spacecraft moving towards the Earth, the beam of light completes one complete circle every 14 s. Calculate the speed of the spacecraft relative to the Earth.

8. A rocket passes two beacons that are at rest relative to the Earth. An astronaut in the rocket measures the time taken for the rocket to travel from the first beacon to the second beacon to be 10·0 s. An observer on Earth measures the time taken for the rocket to travel from the first beacon to the second beacon to be 40·0 s. Calculate the speed of the rocket relative to the Earth.

9. A spacecraft travels to a distant planet at a constant speed relative to the Earth. A clock on the spacecraft records a time of 1 year for the journey while an observer on Earth measures a time of 2 years for the journey. Calculate the speed, in m s 1, of the spacecraft relative to the Earth.

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Relativity – Length contraction

1. Write down the relationship involving the l and contracted length l’ of a moving object observed in two different frames of reference moving at a speed, v, relative to one another (where the proper length is the length measured by an observer at rest with respect to the object and the contracted length is the length measured by another observer moving at a speed, v, relative to the object).

2. In the table shown, use the relativity equation for length contraction to calculate the value of each missing quantity (a) to (f) for an object moving at a constant speed relative to the Earth.

Contracted length Proper length Speed of object / m s 1 (a) 5·00 m 1·00 × 108 (b) 15.0 m 2·00 × 108 0·15 km (c) 2·25 × 108 150 mm (d) 1·04 × 108 30 m 35 m (e) 10 m 11 m (f)

3. A rocket has a length of 20 m when at rest on the Earth. An observer, at rest on the Earth, watches the rocket as it passes at a constant speed of 1·8 × 108 m s 1. Calculate the length of the rocket as measured by the observer.

4. A pi meson is moving at 0·90 c relative to a magnet. The magnet has a length of 2·00 m when at rest to the Earth. Calculate the length of the magnet in the reference frame of the pi meson.

5. In the year 2050 a spacecraft flies over a base station on the Earth. The spacecraft has a speed of 0·8 c. The length of the moving spacecraft is measured as 160 m by a person on the Earth. The spacecraft later lands and the same person measures the length of the now stationary spacecraft. Calculate the length of the stationary spacecraft.

6. A rocket is travelling at 0·50 c relative to a space station. on the rocket measure the length of the space station to be 0.80 km. Calculate the length of the space station according to a technician on the space station.

7. A metre stick has a length of 1·00 m when at rest on the Earth. When in motion relative to an observer on the Earth the same metre stick has a length of 0·50 m. Calculate the speed, in m s 1, of the metre stick.

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8. A spaceship has a length of 220 m when measured at rest on the Earth. The spaceship moves away from the Earth at a constant speed and an observer, on the Earth, now measures its length to be 150 m. Calculate the speed of the spaceship in m s 1.

9. The length of a rocket is measured when at rest and also when moving at a constant speed by an observer at rest relative to the rocket. The observed length is 99·0 % of its length when at rest. Calculate the speed of the rocket.

Relativity questions

1. Two points A and B are separated by 240 m as measured by metre sticks at rest on the Earth. A rocket passes along the line connecting A and B at a constant speed. The time taken for the rocket to travel from A to B, as measured by an observer on the Earth, is 1·00 × 10 6 s.

(a) Show that the speed of the rocket relative to the Earth is 2·40 × 108 m s 1. (b) Calculate the time taken, as measured by a clock in the rocket, for the rocket to travel from A to B. (c) What is the distance between points A and B as measured by metre sticks carried by an observer travelling in the rocket?

2. A spacecraft is travelling at a constant speed of 0·95 c. The spacecraft travels at this speed for 1 year, as measured by a clock on the Earth.

(a) Calculate the time elapsed, in years, as measured by a clock in the spacecraft. (b) Show that the distance travelled by the spacecraft as measured by an observer on the spacecraft is 2·8 × 1015 m. (c) Calculate the distance, in m, the spacecraft will have travelled as measured by an observer on the Earth.

3. A pi meson has a mean lifetime of 2·6 × 10 8 s when at rest. A pi meson moves with a speed of 0·99 c towards the surface of the Earth.

(a) Calculate the mean lifetime of this pi meson as measured by an observer on the Earth. (b) Calculate the mean distance travelled by the pi meson as measured by the observer on the Earth.

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4. A spacecraft moving at 2·4 × 108 m s 1 passes the Earth. An astronaut on the spacecraft finds that it takes 5·0 × 10 7 s for the spacecraft to pass a small marker which is at rest on the Earth.

(a) Calculate the length, in m, of the spacecraft as measured by the astronaut. (b) Calculate the length of the spacecraft as measured by an observer at rest on the Earth.

5. A neon sign flashes with a frequency of 0·2 Hz.

(a) Calculate the time between flashes. (b) An astronaut on a spacecraft passes the Earth at a speed of 0·84 c and sees the neon light flashing. Calculate the time between flashes as observed by the astronaut on the spacecraft.

6. When at rest, a subatomic particle has a lifetime of 0·15 ns. When in motion relative to the Earth the particle’s lifetime is measured by an observer on the Earth as 0·25 ns. Calculate the speed of the particle.

7. A meson is 10·0 km above the Earth’s surface and is moving towards the Earth at a speed of 0·999 c.

(a) Calculate the distance, according to the meson, travelled before it strikes the Earth. (b) Calculate the time taken, according to the meson, for it to travel to the surface of the Earth.

8. The star Alpha Centauri is 4·2 light years away from the Earth. A spacecraft is sent from the Earth to Alpha Centauri. The distance travelled, as measured by the spacecraft, is 3·6 light years.

(a) Calculate the speed of the spacecraft relative to the Earth. (b) Calculate the time taken, in seconds, for the spacecraft to reach Alpha Centauri as measured by an observer on the Earth. (c) Calculate the time taken, in seconds, for the spacecraft to reach Alpha Centauri as measured by a clock on the spacecraft.

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9. Muons, when at rest, have a mean lifetime of 2·60 × 10 8 s. Muons are produced 10 km above the Earth. They move with a speed of 0·995 c towards the surface of the Earth.

(a) Calculate the mean lifetime of the moving muons as measured by an observer on the Earth. (b) Calculate the mean distance travelled by the muons as measured by an observer on the Earth. (c) Calculate the mean distance travelled by the muons as measured by the muons.

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Solutions

Relativity – Fundamental principles

1. (a) 1·0 m s 1 north (b) 2·0 m s 1 south

2. (a) 0·8 m s 1 east (b) 2·8 m s 1 east (c) 2·2 m s 1 west

3. (a) 3·0 m s 1 up (b) 2·0 m s 1 down

4. A = Einstein’s; B = same; C = zero; D = velocity; E = speed of light; F = slow; G = shortened

5. 2000 km h 1

6. (a) No (b) Yes

7. 3 × 108 m s 1

8. (a) 100 s (b) 100 s

9. (a) 3 × 108 m s 1 (b) 3 × 108 m s 1

10. (a) 0·3 × 108 m s 1 (b) 1·5 × 108 m s 1 (c) 1·8 × 108 m s 1 (d) 2·4 × 108 m s 1

11. (a) c (b) 0·67 c (c) 0·5 c (d) 0·33 c

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Relativity – Time dilation

t 1. t ' v2 1 ( ) c2

2. (a) 21·2 h (b) 15·1 year (c) 1043 s (d) 1·32 × 10 4 s (e) 2·10 × 108 m s 1 (f) 9·15 × 107 m s 1

3. (a) 11.11 am (b) 11.11 am

4. 17·1 billion years

5. 9·69 s

6. 3·9 × 10 4 s

7. 2·1 × 108 m s 1 or 0·70 c

8. 2·90 108 m s 1 or 0·97 c

9. 2·60 × 108 m s 1

Relativity – Length contraction

1. l’ = l (1 – v2/c2)

2. (a) 4·71 m (b) 11·2 m (c) 0·227 km (d) 160 mm (e) 1·55 × 108 m s 1 (f) 1·25 × 108 m s 1

3. 16 m

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4. 0·872 m

5. 267 m

6. 0·92 km

7. 2·60 × 108 m s 1

8. 2·19 × 108 m s 1

9. 4·23 × 107 m s 1 or 0.14 c

Relativity questions

1. (b) 0·6 × 10 6 s (c) 144 m

2. (a) 0·31 of a year (c) 8·97 × 1015 m

3. (a) 1·84 × 10 7 s (b) 54·6 m or 54·7 m

4. (a) 120 m (b) 72 m

5. (a) 5 s (b) 9·22 s

6. 0·8 c (2.4× 108 m s 1)

7. (a) 447 m (b) 1·49 × 10 6 s

8. (a) 0·52 c (1.55× 108 m s 1)

(b) 2·55 × 108 s (c) 2·18 × 108 s

9. (a) 2·60 × 10 7 s (b) 77·6 m

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(c) 7·75 m or 7·76 m

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