Hands-On Relativity

Linking Web-Based Resources and Student-Centered Learning

Part A: Using Relativity • Relativity and Momentum: TRIUMF• Relativity and Energy: Fermilab• Relativity and Time: The GPS

Part B: Understanding Relativity • Frames of Reference• Curved Space-Time

http://www.triumf.ca/home/multimedia/videos/speed-lightAvailable in French

http://www.triumf.ca/home/multimedia/videos/la-physique-au-travail

C2.5 analyse the relationships between mass, velocity and momentum

Relativity and Momentum: TRIUMF“Approaching the Speed of Light “

Momentum is calculated using circular acceleration a= v2/r, magnetic force F = qvB,

and Newton’s second law F = ma. 1a) Use these to find a formula for momentum

in terms of the field B and the radius r.

qvB = m v2/rmv = p = qBr

1 b) How is the speed measured?v = d/t

The time interval is measured by using one of the peaks of the histogram.

2) You will be graphing momentum against velocity for some very fast subatomic

particles. What will the graph look like? Give reasons for your prediction.

A) B) C) D) p

v

p

v

p

v

p

v

Should the students be told the right answer at this point or should they just discuss their

ideas?

3) Use the histograms to measure the times and calculate the speeds for the muon – the middle

peak. The distance travelled was 4.40 m.

Momentum (x10-20 kg m/s)

Momentum (MeV/c)

Time (x10-9s)

Velocity (x108 m/s)

Velocity(c)

5.02

7.48

10.7

What units would you use with your class?

Momentum (x10-20 kg m/s)

Momentum (MeV/c)

Time (x10-9s)

Velocity (x108 m/s)

Velocity(c)

5.02 94 21.7 2.03 0.677

7.48 140 18.6 2.36 0.787

10.7 200 16.7 2.63 0.877

4) Graph p (vertical) vs. v for the muons.

5a) What should you graph against p to get a straight line? Hint: It isn’t any power of v.

5d) What is the physical meaning of the slope?

The mass of a muon is 105.7 MeV/c2

You have just shown that p = mv.

Sometimes this is interpreted as mass increases with speed or p = (m)v.

This is the simple version of the activity. You can also;

•choose to use all 24 histograms•analyse the other two particles•examine the sources of error•learn more about these particles•explore the practical uses of the cyclotron that produced these particles

https://www.perimeterinstitute.ca/en/Perimeter_Explorations/Beyond_the_Atom/Beyond_the_Atom_-_Remodelling_Particle_Physics/

C2.7 conduct laboratory inquiries or computer simulations involving explosions in 2 dimensions

Relativity and Energy: Fermilab“Finding the Top Quark”

In 1995 Fermilab discovered evidence for the sixth and final quark of the Standard Model.

This was done by colliding protons and antiprotons to form top-antitop quark pairs.

The quarks decay almost immediately into other particles and can’t be detected directly.

This is a 3-D collision and we will model it with a variety of objects.

We start with a proton and an antiproton moving toward each other with equal high speeds.

The momentum of system is zero, so it must continue to be zero at each stage.

The proton-antiproton pair annihilate and a top-antitop pair appear, moving very slowly.

Which way will they move?A)in the same directions as the original pairB)opposite the proton-antiproton velocitiesC)opposite each otherD)There is not enough information Momentum must be conserved.They must move opposite each other.

Next the top-antitop pair can broke into 4 jets of particles and a muon.

Which ways will the five momenta point?

Any directions that conserve momentum.

They can all move out in a 2-D plane. This is unlikely, but it did happen!

This is the standard

When you add the five momenta you should find that they add to

A) around 300 GeVB) around 0 GeVC) exactly 0 GeVD) a different number each time

C) exactly 0 GeVconservation of pB) around 0 GeV experimental error

Use a scale of 1 mm to 1 GeV/c and measure the horizontal and vertical components of each

of the momenta. Find the total momentum.

Horizontal(mm)

-94

Vertical(mm)

-15

Use a scale of 1mm to 1 GeV/c measure the horizontal and vertical components of each of the 5 momenta and find the total momentum.

Horizontal(mm)

-94 -32 11 58 22 -35

Vertical(mm)

-15 44 14 20 -62 1

The momentum does not add to zero. Why?

Neutrinos are not detected.

Use conservation of momentum to find the momentum of the missing neutrino and

draw this momentum on the event display.

This is the standard

The equation E = mc2 is for a particle at rest. The full equation is E2 = (pc) 2 + (mc2) 2.

The momenta of these particles is so large that we can ignore the (mc2) 2 term.

E2 = (pc) 2 + (mc2) 2 The equation can be simplified to E = pc.

This simplification, E = pc, means that a fast particle with 95.5 GeV/c of momentum has

about 95.5 GeV of energy.

Find the total energy of the particles - including the neutrino - by adding their energies.

This energy produced an almost stationary pair of top-antitop quarks.

What is the mass of the top quark?A) 330 GeV B) 115 GeVC) 295 GeV D) 147 GeV

The energy of all the particles, including the neutrino, produced two quarks.

The mass of one is 115 GeV.

You may prefer to use GeV/c 2 for units.

Fermilab collided a proton and an antiproton together in order to find the top quark. You

have a friend studying biology who thinks that this is a rather sloppy way to dissect protons.

Explain how this collision is not like a dissection. (Hint: The mass of a proton is 1 GeV.)

The top quarks (m = 330 GeV) were not inside the protons. Particle physics is more like magic

and alchemy than chemistry.

There is much more to do and learn in

Beyond the Atom: Remodelling Particle Physics

You can order this and other free resources online or at the Perimeter Institute table.

https://www.perimeterinstitute.ca/Perimeter_Inspirations/GPS_%26_Relativity/GPS_%26_Relativity/ Available in French

A 1.13 express the results of any calculations involving data accurately and precisely, to the appropriate

number of decimal places or significant figures

Relativity and Time: The GPS“Everyday Einstein”

In special relativity, gamma is a measure of how different things are compared to classical situations. What is gamma at

really slow, non-relativistic speeds?

A)undefined B) infinite C) zero D) oneD) one

The GPS satellites move at 3.874 x 103 m/s. The speed of light is 2.997 924 58 x 108 m/s.

Calculate gamma for the satellites.

Most calculators give 1.000000000.

When v is much less than c, the relationship is well approximated by

Calculate gamma.Most calculators give 1.000000000.

Just calculate the second term – the error that would result if you ignored relativity.

What should you enter into your calculator?

You only have 4 digits for the satellite speed.

Powers of ten should be treated separately.

The correct answer isA) 8.34939395 x 10-11 sB) 8.349 x 10-11 sC) 8.35 x 10-11 sD)10-10 s

8.349 x 10-11 s is correct, but unnecessary.We only need the order of magnitude to determine where our digits stop being significant, so 10-10 is all that is needed.

If 1.000 000 000 0 seconds pass on the satellite, 1.000 000 000 1 seconds pass on the Earth.

Why does this matter?

A) It doesn`t, 3874 km/s is not a relativistic speed. B) This time error is multiplied by c.C) The error is cumulative as time passes.

10-10 s x 3 x 108 m/s = 3 cm. Not significant.However, there are 24 x 60 x 60 seconds in a day. After a day your position will be wrong by 2 km.

B3.1 distinguish between reference systems (inertial and non-inertial) with respect to real

and apparent forces acting within such systems.

Frames of Reference“Everyday Einstein”

A bottle with water has a small hole in the top. It is turned upside down and dropped.

What happens to the water as the bottle falls?

A)It pours out as if the bottle was stationary.B)It pours out slower than normal. C) It pours out faster than normal.D) It stays in the bottle.

The bottle is thrown upwards and the hole is uncovered. What happens to the water

while the bottle is in the air?

A)It pours out on the way up and down.B)It stays in the bottle.C)It pours out only on the way up.D)It pours out only on the way down.

A cup of water is on a tray which is swung in a horizontal circle. Why does the water stay in

the cup and the cup stay on the tray?

Explain with a FBD in the frame of reference of the Earth and then for the tray

In both frames there is a normal force perpendicular to the tray.

In the Earth Frame this force makes everything move in circular motion (and

cancels a bit of gravity). In the tray frame this force is balanced by the

‘artificial’ gravity (and a bit of real gravity).

The water stays in the cup and the cup stays on the tray because there is an acceleration _______________ which is equivalent to a

gravitational field ________________.

A) inwards, outwards.B) outwards , outwards.C) inwards, inwards.D) outwards, inwards

In free-fall you feel no gravity.

An accelerating frame has an artificial gravity.

Fictitious forces appear and disappear when you change reference frames.

Fictitious forces produce equal accelerations for all masses.

Is gravity a fictitious force?

When Einstein realized that gravity and accelerating frames were equivalent, he said

it was the ‘happiest thought of his life’.

Next, he explored what effect gravity would have on time. You will do the same.

Bob is at the rear and sends pulses of light to Alice at the front every 100 ns.

How often does Alice receive the pulses?A) every 100 ns B) more frequently

C) less frequentlyA) You can’t tell if your frame is moving.

Fig. 2: A rocket moving with constant acceleration.Fig. 1: A rocket moving with constant velocity.

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On the left are the positions of a rocket moving up at a constant velocity and shown every 100 ns.

On the right are the positions of the rocket when it is accelerating upwards.

How often does Alice receive the pulses now?A)every 100 ns B) more frequently

C) less frequentlyC) The light must cover extra distance.

Fig. 2: A rocket moving with constant acceleration.Fig. 1: A rocket moving with constant velocity.

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12 3 3

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The rocket is stationary in a gravitational field. How often does Alice receive the pulses? A)every 100 ns B) more frequently

C) less frequentlyHint: Will this situation resemble an accelerating

rocket or one travelling at constant velocity?

The signals will arrive less frequently. Alice will say that Bob’s clock is running slowly.

Suppose Alice sends signals every 100 ns to Bob? How often does Bob receive the pulses?

A) every 100 ns B) more frequentlyC) less frequently

Hint: Alice received the signals less frequently because she was moving away from the

pulses. However, Bob is moving toward them.Bob receives them more frequently and says that

Alice’s clock is moving fast.

Einstein’s equivalence principle - between gravity and acceleration - predicts that

gravity will slow time.

That was the easy part. It took Einstein another 10 years to figure out the rest and

he needed lots of help with the math.

We will explore it using models not math.

“Revolutions in Science” “Dark Matter: Bonus Materials”

https://www.perimeterinstitute.ca/Perimeter_Inspirations/Revolutions_in_Science/Revolutions_in_Science/ http://www.perimeterinstitute.ca/Perimeter_Explorations/The_Mystery_of_Dark_Matter/The_Mystery_of_Dark_Matter/

D3.1 identify, and compare the properties of fundamental forces that are associated with

different theories and models of physics (e.g. the theory of general relativity)

Curved Space-Time:

Alice and Bob in Wonderlandhttp://www.perimeterinstitute.ca/en/Outreach/Alice_and_Bob_in_Wonderland/Alice_and_Bob_in_Wonderland/

How can the ground always be accelerating?

We need to consider warped spacetime.

Alice falls from a ladder in flat space.

`

Place a piece of masking tape along Bob’s path and another for Jane.

Jane’s tape is wrinkled downward.

Now try it on a beach ball.

Jane’s path is smooth.Bob’s path is wrinkled upward.

What if Alice stays on top of the ladder?

Alice’s path is a different length from Bob’s. Her time and Bob’s are different.

Curved spacetime is a nice idea. What evidence is there that it is true?

Trace an orbit on paper and on a beach ball.

The orbits of the planets precess like this.

What will happen to light as it passes near the curved space of a massive body?

Place masking tape smoothly toward an upturned bowl, onto the bowl’s side and off.

Curved space bends light.

What will happen if the bowl is right-side up?

The 2-D bowl space curve into a 3rd dimension. Our 3-D space curves into a fourth.

You can see what the effects will look like by placing a wine glass base over a spot.

The arcs and circles below show how light is bent by gravitational lensing.

Gravitational lenses are used by astronomers to help them see extremely

distant objects and to detect dark matter.

Alice and Bob in Wonderlandhttp://www.perimeterinstitute.ca/en/Outreach/

Alice_and_Bob_in_Wonderland/Alice_and_Bob_in_Wonderland/

Curved spacetime is needed to understand what black holes are like.

This is an ordinary black hole. You can only see it from one direction.

A black hole look likes this from all directions.

• Al`s relativistic adventures: http://www.onestick.com/relativity/

A simple yet accurate picture book of special relativity

• Einstein Online: http://www.einstein-online.info/ A huge source of relativity from simple to very complicated

• Spacetime Diagrams: http://roberta.tevlin.ca/Rel/ST%20I/Spacetime%20I.htm

A wonderful tool to visualize relativity. Available in French.

Other Useful Sites

These activities and others are gathered into ten lessons, which you can find at http://roberta.tevlin.ca/Workshops.htm

Please feel free to contact me with questions and suggestions.

roberta@tevlin.ca

Thank you