IB PHYSICS OPTION: RELATIVITY • Atit Bhargava • Brian ... · This book provides questions and...

IB PHYSICS OPTION: RELATIVITY

• Atit Bhargava • Brian Shadwick •

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Science Press. ABN 98 000 073 861

© Science Press 2016First published 2016

Science PressPrivate Bag 7023 Marrickville NSW 1475 AustraliaTel: +61 2 9516 1122 Fax: +61 2 9550 [email protected] www.sciencepress.com.au

Contents

Words to Watch iv

Introduction v

Dot Points

The Beginnings of Relativity vi

Lorentz Transformations vi

Spacetime Diagrams vi

Relativistic Mechanics vi

General Relativity vi

Questions

The Beginnings of Relativity 1

Lorentz Transformations 9

Spacetime Diagrams 27

Relativistic Mechanics 33

General Relativity 41

Answers

The Beginnings of Relativity 46

Lorentz Transformations 49

Spacetime Diagrams 57

Relativistic Mechanics 59

General Relativity 61

Appendices

Data Sheet 62

Periodic Table 63

iii Contents

Science Press

Dot Point IB Physics Option: Relativity

account, account for State reasons for, report on,

give an account of, narrate a series of events or

transactions.

analyse Interpret data to reach conclusions.

annotate Add brief notes to a diagram or graph.

apply Put to use in a particular situation.

assess Make a judgement about the value of

something.

calculate Find a numerical answer.

clarify Make clear or plain.

classify Arrange into classes, groups or categories.

comment Give a judgement based on a given

statement or result of a calculation.

compare Estimate, measure or note how things are

similar or different.

construct Represent or develop in graphical form.

contrast Show how things are different or opposite.

create Originate or bring into existence.

deduce Reach a conclusion from given information.

define Give the precise meaning of a word, phrase or

physical quantity.

demonstrate Show by example.

derive Manipulate a mathematical relationship(s) to

give a new equation or relationship.

describe Give a detailed account.

design Produce a plan, simulation or model.

determine Find the only possible answer.

discuss Talk or write about a topic, taking into account

different issues or ideas.

distinguish Give differences between two or more

different items.

draw Represent by means of pencil lines.

estimate Find an approximate value for an unknown

quantity.

evaluate Assess the implications and limitations.

examine Inquire into.

explain Make something clear or easy to understand.

extract Choose relevant and/or appropriate details.

extrapolate Infer from what is known.

hypothesise Suggest an explanation for a group of facts or phenomena.

identify Recognise and name.

interpret Draw meaning from.

investigate Plan, inquire into and draw conclusions about.

justify Support an argument or conclusion.

label Add labels to a diagram.

list Give a sequence of names or other brief answers.

measure Find a value for a quantity.

outline Give a brief account or summary.

plan Use strategies to develop a series of steps or processes.

predict Give an expected result.

propose Put forward a plan or suggestion for consideration or action.

recall Present remembered ideas, facts or experiences.

relate Tell or report about happenings, events or circumstances.

represent Use words, images or symbols to convey meaning.

select Choose in preference to another or others.

sequence Arrange in order.

show Give the steps in a calculation or derivation.

sketch Make a quick, rough drawing of something.

solve Work out the answer to a problem.

state Give a specific name, value or other brief answer.

suggest Put forward an idea for consideration.

summarise Give a brief statement of the main points.

synthesise Combine various elements to make a whole.

Words to Watch

Dot Point IB Physics Option: Relativityiv

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Words to Watch

What the book includes

This book provides questions and answers for each dot point in the IB Physics Option: Relativity syllabus from the International Baccalaureate Diploma Programme for Physics:

• The Beginnings of Relativity

• Lorentz Transformations

• Spacetime Diagrams

• Relativistic Mechanics

• General Relativity

Format of the book

The book has been formatted in the following way:

1.1 Subtopic from syllabus.

1.1.1 Assessment statement from syllabus.

1.1.1.1 First question for this assessment statement.

1.1.1.2 Second question for this assessment statement.

The number of lines provided for each answer gives an indication of how many marks the question might be worth in an examination. As a rough rule, every two lines of answer might be worth 1 mark.

How to use the book

Completing all questions will provide you with a summary of all the work you need to know from the syllabus. You may have done work in addition to this with your teacher as extension work. Obviously this is not covered, but you may need to know this additional work for your school exams.

When working through the questions, write the answers you have to look up in a different colour to those you know without having to research the work. This will provide you with a quick reference for work needing further revision.

Introduction

v

Science Press

Dot Point IB Physics Option: Relativity Introduction

RelativityA1 The Beginnings of Relativity 2

A.1.1 Reference frames. 2A.1.2 Galilean relativity and Newton’s postulates concerning time and space. 4A.1.3 Maxwell and the constancy of the speed of light. 7A.1.4 Forces on a charge or current. 7

Answers to The Beginnings of Relativity 46

A2 Lorentz Transformations 10

A.2.1 The two postulates of special relativity. 10A.2.2 Clock synchronisation. 14A.2.3 The Lorentz transformations. 14A.2.4 Velocity addition. 16A.2.5 Invariant quantities (spacetime interval, proper time, proper length and rest mass). 17A.2.6 Time dilation. 18A.2.7 Length contraction. 20A.2.8 The muon decay experiment. 26

Answers to Lorentz Transformations 49

A3 Spacetime Diagrams 28

A.3.1 Spacetime diagrams. 28A.3.2 Worldlines. 28A.3.3 The twin paradox. 32

Answers to Spacetime Diagrams 57

A4 Relativistic Mechanics 34

A.4.1 Total energy and rest energy. 34A.4.2 Relativistic momentum. 37A.4.3 Particle acceleration. 38A.4.4 Electric charge as an invariant quantity. 38A.4.5 Photons. 39A.4.6 MeV c–2 as the unit of mass and MeV c–1 as the unit of momentum. 37

Answers to Relativistic Mechanics 59

A5 General Relativity 42

A.5.1 The equivalence principle. 42A.5.2 The bending of light. 42A.5.3 Gravitational redshift and the Pound-Rebka-Snider experiment. 42A.5.4 Schwarzschild black holes. 43A.5.5 Event horizons. 43A.5.6 Time dilation near a black hole. 43A.5.7 Applications of general relativity to the Universe as a whole. 44

Answers to General Relativity 61

Dot Points

Dot Point IB Physics Option: RelativityviDot Points

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DOT POINT

The Beginnings of Relativity

A1

1 The Beginnings of Relativity

Science Press


A1 The Beginnings of Relativity.

A.1.1 Reference frames.

A.1.1.1 Identify an everyday example of a frame of reference. ��

��

A.1.1.2 Using an appropriate example, discuss why it is important for us to consider our frame of reference when we make measurements.

��

��

A.1.1.3 You regain consciousness some time after an asteroid hit your spaceship. You are unaware of any movement of the craft. You wonder if you are still on course and moving towards Andromeda galaxy. Suddenly a comet shoots past you, seemingly parallel to your path and moving straight ahead.

Which of the following interpretations of this observation is not possible?

(A) Both you and the comet are travelling towards Andromeda, but the comet is moving faster than you.

(B) You are stationary and the comet is moving past you towards Andromeda.

(C) You are moving backwards and the comet is moving towards Andromeda.

(D) You are moving towards Andromeda and the comet is moving away from Andromeda.

Extension: Give two other possible interpretations of this observation. ��

��

Use the following information to answer the next TWO questions.

A.1.1.4 A boy drops a ball out of the window of a car that is accelerating away from a set of traffic lights.

Which diagram best represents the path taken by the ball as seen by the boy in the car?

Direction of acceleration of car

(A) (B) (C) (D)

Extension: Explain your answer. ��

A.1.1.5 Which diagram represents the motion of the ball as observed by a stationary person on the footpath?

(A) A (B) B (C) C (D) D


A.1.1.6 What do we mean by ‘classical physics’? ��

��

A.1.1.7 What is the major limitation of classical physics? ��

��

2The Beginnings of Relativity

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A.1.1.8 In classical physics, what is meant by the phrase ‘no absolute zero velocity’? ��

��

A.1.1.9 Recall the classical principle of relativity. ��

��

A.1.1.10 By using a suitable example, clarify the concept of a frame of reference. ��

��

A.1.1.11 Define, giving appropriate examples, an inertial frame of reference. ��

��

A.1.1.12 Define, giving appropriate examples, a non-inertial frame of reference. ��

��

A.1.1.13 An astronaut in a spaceship tied her mascot to a string and hung it from the ceiling. One day she noticed, that instead of hanging straight down, it hung at an angle.

(a) Account for this. ��

(b) Identify the frame of reference the mascot is in when it hangs straight down. Justify your answer.

��

(c) Identify the frame of reference of the spaceship when the mascot hangs at an angle. Justify your answer.

��

A.1.1.14 Which of the following is an inertial frame of reference?

(A) A car turning a corner.

(B) A rocket during launch.

(C) A rocket moving at constant velocity in outer space.

(D) A bus which is slowing down.

Extension: Which choice is a non-inertial frame of reference? Why? ��

A.1.1.15 What best summarises the characteristics of an inertial frame of reference?

(A) It is stationary. (B) It is moving with constant velocity.

(C) It is not turning a corner. (D) It is accelerating.

Extension: What is the characteristic of a non-inertial frame of reference? ��

A.1.1.16 The diagrams show tennis balls hanging on light strings from the ceilings of three different vehicles X, Y and Z.

Which vehicle(s) is/are inertial frames of reference?

(A) X only. (B) Y only. (C) Z only. (D) Y and Z only.


A.1.1.17 Which of the following is a correct statement?

(A) A net force cannot exist in an inertial frame of reference.

(B) A net force cannot exist inside a non-inertial frame of reference.

(C) An inertial frame of reference can be detected by an observer inside the system.

(D) A non-inertial frame of reference cannot be detected by an observer inside the system.


X Y Z


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A.1.2 Galilean relativity and Newton’s postulates concerning time and space.

Students are advised to study section A.2.1 along with this section.

A.1.2.1 What do what we mean by a Galilean transformation? ��

��

A.1.2.2 In what situations do Galilean transformations not hold? Explain your answer. ��

��

A.1.2.3 Relate the four coordinates of spacetime (x, y, z and t) in an inertial reference frame moving with a speed v in direction x relative to another reference frame (x′, y′, z′ and t′).

��

A.1.2.4 What assumption do Galilean equations make which was proven untrue by Einstein? ��

��

A.1.2.5 A student in a bus which is moving at constant velocity from the left to the right throws a ball straight up into the air. The path the ball follows is observed by Michael inside the bus and by Christian outside the bus. Which diagrams show what each sees?

Extension: What type of frame of reference is each observer in?

��

A.1.2.6 Object X is moving east at 30 m s–1. Object Y is moving west at 25 m s–1. Object Z is moving east at 15 m s–1. Calculate the velocity of:

(a) X relative to Y. ��

(b) X relative to Z. ��

(c) Y relative to X. ��

(d) Y relative to Z. ��

(e) Z relative to X. ��

(f) Z relative to Y. ��

A.1.2.7 From your answers above, identify the relationship between the velocity of object X relative to object Y and the velocity of object Y relative to object X.

��

A.1.2.8 Object X is moving east at 24 m s–1. Object Y is moving north at 18 m s–1. Calculate the velocity of the following.

(a) X relative to Y. ��

(b) Y relative to X. ��

A.1.2.9 A person can row a boat at 1.75 m s–1 in still water. He rows on a river which flows at a speed of 0.45 m s–1.

Calculate the velocity of the person relative to the banks of the river if he rows:

(a) With the flow. ��

(b) Against the flow. ��

Michael sees Christian sees

(A)

(B)

(C)

(D)


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Calculate the velocity of the boat relative to the water if he rows:

(c) With the flow. ��

(d) Against the flow. ��

A.1.2.10 In an experiment, Pat and Chris took measurements to determine how fast a moving footway was moving.

They took two sets of readings: firstly with Chris walking on the moving footway in the same direction as the footway movement, and secondly with Chris walking at the same rate in the opposite direction. In each case, Pat timed how long it took Chris to walk 50 metres. The results are shown in the table.

Use these results to answer the following questions.

(a) Calculate how fast the footway was moving. ��

(b) Calculate how fast Chris was walking. ��

(c) Calculate Chris’s speed relative to the footway in each case. ��

(d) Calculate Chris’s speed relative to Pat in each case. ��

A.1.2.11 The diagram shows several cars approaching an intersection.

Use the information in the diagram to complete the table.

H I J K L

H ��

I ��

J ��

K ��

L ��

Velocity of ...

Relative to ...

Time to walk 50 m in same direction as footway

movement (s)

Time to walk 50 m in opposite direction to footway

movement (s)

7.1 17.2

7.0 16.6

6.8 16.0

6.8 17.5

7.5 16.2

6.7 17.2

7.7 16.0

(40 km h–1) (30 km h–1)

(40 km h–1)

(50 km h–1)(50 km h–1)

LI

J

K

H

N

60°


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A.1.2.12 A pion is an unstable particle that decays into two photons. A particular pion, travelling at 0.950 c with respect to an observer at rest, decays into two photons, X and Y travelling in opposite directions as shown in the diagram.

Pion

0.950 cX Y

Photons after decay

The speed of both photons as measured by the observer at rest with respect to the pion is c.

(a) Calculate the velocity of photon Y with respect to the observer using Galilean kinematics.

��

(b) Calculate the velocity of photon Y with respect to the observer using relativistic kinematics.

��

A.1.2.13 Two carts are on parallel, horizontal tracks next to each other. Tom is an observer in cart T and Dick is an observer in cart D. Both Tom and Dick consider themselves to be at the origin of their frame of reference and for each, their x-axis is parallel to the tracks. Dick’s cart is moving with constant velocity v towards the right and is carrying a slab of black marble.

The diagram shows the relative positions of the two carts at time zero, and after time t seconds. Notice that Dick’s cart has moved a distance vt ahead of Tom’s cart.

Cart T

Cart D

Tom’s cart at time 0

Dick’s cart at time 0

Tom’s cart at time t

Dick’s cart at time tVt

x1¢ x2¢

x1 x2

At time t, Tom measures the ends of the marble slab to be at x1′ and x2′ from his frame of reference while Dick measures them to be at x1 and x2 from his frame of reference.

(a) Use a Gallilean transformation to show that both Tom and Dick measure the slab of marble to be the same length.

��

(b) State the relationship between (x2′ – x1′) and (x2 – x1) using a relativistic transformation.

��

(c) With reference to special relativity, state why it is important that the measurements of the positions of the ends of the marble are made simultaneously.

��

(d) On Dick’s cart there is a clock which he turns on then off and notices it records 1.3 s. Tom watches the clock as it is turned on and off by Tom but only records a time of 1.6 s.

(i) Who records the proper time? Explain why. ��

(ii) How fast is Dick’s cart travelling? ��


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DOT POINT

Answers

45 Answers

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The Beginnings of Relativity

A.1.1.1 The idea of a frame of reference refers to the environment in which you make measurements. A common frame of reference would be your school laboratory.

A.1.1.2 Answers will vary, for example: Measurements made without considering the frame of reference in which they are made can be misleading. For example, ancient astronomers considered the Earth to be stationary and that the Sun and Moon (and the rest of the Universe) rotated about it. This misconception guided the thinking of those astronomers for centuries before improved technology enabled later astronomers to discover the true situation.

A.1.1.3 D The comet is stationary and you are moving away from Andromeda. You are both moving away from Andromeda, but you are moving away much faster than the comet.

A.1.1.4 B Because the car is accelerating, the motion of the ball relative to the car will involve uniform acceleration both horizontally

and vertically. From D, r = ut + 12 at2 the ratio of the displacement of the ball in each direction will be 1 : 3 : 5 : 7 ... This will

result in relative motion in a straight line relative to the car and the boy inside it.

A.1.1.5 C Ball will accelerate downwards but travel horizontally with the constant speed it has when released, therefore its path will be that of a typical projectile.

A.1.1.6 ‘Classical physics’ refers to the laws of physics developed before the discovery of quantum theory.

A.1.1.7 Classical physics describes the properties of particles larger than atomic size, but does not completely describe those of particles smaller than atoms.

A.1.1.8 While we regard ‘non-moving’ objects in our frame of reference as having zero velocity, this disregards the fact that our frame of reference is most certainly moving. While your classroom is stationary, it is actually moving with the rotation of the Earth, the orbital speed of the Earth, the motion of the Milky Way Galaxy through space, the expansion of the Universe, and perhaps the relative motion of multiple universes. If we consider this, then there may well be no ‘absolute zero velocity’.

A.1.1.9 All motion is relative might be a simple statement, but at this level you should be a little more specific, for example: All steady (stationary or constant velocity) motion is relative and cannot be detected without reference to a point outside the frame of reference.

A.1.1.10 A frame of reference simply refers to you and the objects around you that are in the same state of motion as you. For example, if you are in a car driving along the road, then everything inside the car, including the car itself, is your frame of reference. Everything outside the car is in a different frame of reference.

A.1.1.11 A frame of reference that is moving with constant motion (stationary or constant velocity) is known as an inertial frame of reference. A spaceship at constant velocity in deep space would be an inertial frame of reference. An object in a stable circular orbit is also classified as an inertial frame of reference. Motion cannot be detected in an inertial frame of reference.

A.1.1.12 A frame of reference that is accelerating is known as a non-inertial frame of reference. A plane accelerating down the runway for take-off, a car turning a corner, or a braking train are non-inertial frames of reference. Motion is detectable in a non-inertial frame of reference.

A.1.1.13 (a) Spaceship was no longer an inertial frame of reference. It was accelerating in the opposite direction to the angle of hang.

(b) Inertial frame of reference. If the spaceship was accelerating, inertial forces would be noticeable (the mascot would not hang vertically down). (Note: We cannot determine by any experiment if the frame of reference is stationary or moving with constant velocity without referring to some point outside the frame of reference.)

(c) Non-inertial – accelerated motion is detectable because of the inertial forces acting on the mascot and causing it to hang at an angle.

A.1.1.14 C All the others are accelerating frames of reference.

A.1.1.15 B This is the best answer as it covers both stationary and constant velocity.

A.1.1.16 D Y and Z both show the ball hanging at an angle, so there must be an inertial force acting on the balls – the buses must be accelerating, and therefore non-inertial frames of reference. X shows no inertial forces acting on the hanging object, therefore an inertial frame of reference.

A.1.1.17 C While an observer inside an inertial frame of reference cannot detect whether the system is moving or not, the absence of any inertial forces acting on suspended objects will indicate that the system is not accelerating and is therefore an inertial frame of reference.

A.1.2.1 A Galilean transformation is one in which normal mathematics can be applied in order to determine the relative motion. It compares motion in different frames of reference using only Newtonian physics and ignores relativistic effects.

A.1.2.2 Galilean transformations hold at low speeds where relativistic effects can be ignored. If objects are moving at significant proportions of the speed of light then relativistic effects must be considered, so a simple Galilean transformation would be invalid.

A.1.2.3 Since y and z are perpendicular to the direction of motion:

x′ = x – vt

t′ = t

46Answers

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