Key Areas covered The speed of light in a vacuum is the same for all observers. The constancy of the speed of light led Einstein to postulate that measurements of space and time for a moving observer are changed relative to those for a stationary observer. Length contraction and time dilation.

What we will do today State what is meant by time dilation Carry out calculations on the above State what is meant by length contraction Carry out calculations on the above Discuss why we can’t see time dilation in practice in our everyday life. Discuss what proof we have of time dilation

Special Relativity

Lesson starter A train is travelling at 200mph. A man stands at the back of the train with a gun. He fires it towards the man at the front of the train. If the bullet has a velocity of only 100mph, will it hit the man at the front of the train?

It’s all relative!!!! To the men on the train, it’s as if they are stationary, the bullet will kill man. If you were an eye witness, you would see the train moving at 200mph as well as everything on the train (the murderer and the gun included). Therefore the bullet would appear to travel at 300mph ( ).

But what if…. That’s all well and good for low speeds but what would happen if the train was travelling at the speed of light? Would you, as a bystander, see the bullet travelling faster than the speed of light? As we know, nothing travels faster than the speed of light. So what would happen? Theory of relativity explained in 7 mins – YouTube Theory of relativity explained in 7 mins – YouTube (watch parts 1 and 2 only) Or for a more detailed explanation Al's Relativistic Adventures

Time dilation Imagine a lamp which sends a pulse of light at the same time as producing a click. The light is reflected from a mirror, at a known distance, D, from the lamp. When it arrives back at the lamp it produces a second click. The total time will be: t = 2D c (c – speed of light)

Time dilation Now imagine that the two lamps are moving at an identical horizontal velocity. To an observer moving with the lamps nothing will have changed. However, if there is a stationary observer watching the lamps move he will see the pulses of light take a different path and move a longer distance, 2h.

Time dilation The time between clicks in this case will be: t= 2h c Therefore, time will be different for two observers watching an identical system (as h is clearly bigger than D).

Time dilation animation showing time dilation at different velocitiesanimation showing time dilation at different velocities

What is meant by time dilation? The time observed in a moving system will always be greater than that measured in the stationary frame of reference. Time dilation is the difference in a time interval as measured by a stationary observer and a moving observer. ie a stationary observer will record a greater time than a moving observer for the same journey travelling at speeds close to the speed of light.

Equation for time dilation t’= t. √1 – v 2 c2c2 t’ = time reference for the stationary observer t = time reference for the moving observer v = velocity of moving observer c = 3 x 10 8 ms -1 NB: v is often given as a unit of c i.e. 0.7c (so 0.7 x 3x10 8 = 2.1x10 8 ). In this case you can leave v = 0.7 and c = 1

Example 1 A spacecraft leaves Earth and travels at a constant speed of 0.6c to its destination. An astronaut on board records a flight time of 5 days. Calculate the time taken for the journey as measured by an observer on Earth. t’= t. √1 – v 2 c 2 t’ =5. √1 – t’ = 6.25 days

Example 2 A rocket leaves a planet and travels at a constant speed of 0.8c to a destination. An observer on the planet records a time of 20h. Calculate the time taken for the journey as measured by the astronaut on board. t’= t. √1 – v 2 c 2 20 =t. √1 – x √1 – = t 1 2 t = 20 x (0.6) t = 12 h

2012 Revised Higher C

2014 Revised Higher B

CfE Higher Specimen Paper

Marking instructions

CfE Higher 2015

Questions Activity sheets: Special relativity and time dilation You should now be able to answer all questions in class jotter

Answers – Special Relativity 1. (a) 1·0 ms −1 north (b) 2·0 ms −1 south 2. (a) 0·8 ms −1 east (b) 2·8 ms −1 east (c) 2·2 ms −1 west 3. (a) 3·0 ms −1 (b) 2·0 ms −1 4. A = Einstein’s; B = same; C = zero; D = velocity; E = speed of light; F = slow; G = shortened kmh −1 6. (a) No (b) Yes 7. 3 × 10 8 ms −1 8. (a) 100 s (b) 100 s 9. (a) 3 × 10 8 ms −1 (b) 3 × 10 8 ms −1 10. (a) 0·3 × 10 8 ms −1 (b) 1·5 × 10 8 ms −1 (c) 1·8 × 10 8 ms −1 (d) 2·4 × 10 8 ms −1 11. (a) c (b) 0·67 c (c) 0·5 c (d) 0·33 c

Answers – time dilation 1. Teacher Check 2. (a) 21·2 h (b) 15·1 year (c) 1043 s (d) 1·32 × 10 −4 s (e) 2·10 × 10 8 ms −1 (f) 9·15 × 10 7 ms −1 3. (a) am (b) am 4. 17·1 billion years 5. 9·69 s 6. 3·9 × 10 −4 s 7. 2·1 × 10 8 ms −1 8. 2· ms −1 9. 2·60 × 10 8 ms −1

What proof do we have for time dilation? Well there is experimental evidence to support Einstein’s theory. There is a particle known as a muon that is created in the upper atmosphere. Muons only exist for a short time, they have a half- life of 1.56 x s. This means that for every million muons created at a height of 10km only 0.3 should reach the surface of the Earth. In actual fact around 5000 are detected. This is because the ‘muon clock’ runs slowly compared to the observer on Earth and the muon reaches the ground.

Past Paper Qu Specimen paper Qu: 4

Length Contraction Theory of relativity explained in 7 mins – YouTube Now watch part 3 onwards

Length contraction A similar effect occurs with the length of a moving object. This is know as length contraction. It can be defined as follows: the decrease in length of an object moving relative to an observer. l’= l √1 – v 2 c 2 l’ = length for the stationary observer l = length for the moving observer

General rule When moving at velocities close to the speed of light, for the stationary observer (standing watching): 1.Time is longer 2.The length is shorter When compared to the moving observer (onboard the space craft).

Example 1 An observer on Earth sees a spaceship travelling at 0.7c. If the rocket is measured to be 36m in length when at rest on Earth, how long is the moving rocket ship as measured by the observer on Earth? l’= l √1 – v 2 c 2 l’= 36 √1 – l’= 36 x (0.714) l’ = 25.7m

An observer on Earth sees a rocket zoom by at 0.95c. If the rocket is measured to be 5.5m in length, how long is the rocket ship as measured by the astronaut inside the rocket? l’= l √1 – v 2 c 2 5.5= l √1 – = l x (0.312) l = 5.5 / l = 17.6m Example 2

2012 Revised Higher B

Revised Higher specimen paper A

2015 CfE Higher

Questions Activity sheets: Length dilation You should now be able to answer all questions in class jotter

Answers – length contraction 1. Teacher Check 2. (a) 4·71 m (b) 11·2 m (c) 0·227 km (d) 160 mm (e) 1·55 × 10 8 ms −1 (f) 1·25 × 10 8 ms − m 4. 0·872 m m 6. 0·92 km 7. 2·60 × 10 8 ms −1 8. 2·19 × 10 8 ms −1 9. 4·23 × 10 7 ms −1

Travelling at light speed Ever wondered what you would see if you travelled at light speed? virtual tour of a city at light speed showing length contraction

Why do we not notice time dilation when we are moving at speeds NOT close to c? This can be explained using the Lorentz factor. The Lorentz factor appears in special relativity equations for both time dilation and length contraction. It has the symbol γ and can be expressed as: γ = 1. √1- (v/c) 2

Why do we not notice time dilation when we are moving at speeds NOT close to c? Therefore we can see that if the velocity is much less than c, there will hardly be any effect. For example, the max velocity of a Boeing 747 is 270 ms -1, using the Lorentz factor: γ = 1. √1- (v/c) 2 γ = 1 And therefore has no effect Only velocities close to the speed of light have an effect.

Why do we not notice time dilation when we are moving at speeds NOT close to c? Velocities much less than the speed of light correspond to a Lorentz factor of approximately 1. Therefore there is negligible change of length / time observed.

Questions Activity sheets: Relativity Miscellaneous You should now be able to answer all questions in class jotter

Answers – Relativity Miscellaneous 1. (a) Teacher Check (b) 1·67 × 10 −6 s (c) 144 m 2. (a) 0·31 of a year (b) Teacher Check (c) 8·97 × m 3. (a) 1·84 × 10 −7 s (b) 54·6 m or 54·7 m m 4. 0·872 m m 6. 0·92 km 7. 2·60 × 10 8 ms −1 8. 2·19 × 10 8 ms −1 9. 4·23 × 10 7 ms −1 or 0.14 c

2013 Revised Higher (1/2)

2013 Revised Higher (2/2)

Solution