Hands-On Relativity

1. Hands-On Relativity Linking Web-Based Resources and Student-Centered Learning

2. 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

3. Relativity and Momentum: TRIUMF“Approaching the Speed of Light “ http://www.triumf.ca/home/multimedia/videos/speed-light Available in French http://www.triumf.ca/home/multimedia/videos/la-physique-au-travail C2.5 analyse the relationships between mass, velocity and momentum

4. Momentum is calculated using circular accelerationa= 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/r mv = p = qBr

5. 1 b) How is the speed measured? v = Dd/Dt The time interval is measured by using one of the peaks of the histogram.

6. p p p p v v v v 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) Should the students be told the right answer at this point or should they just discuss their ideas?

7. 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. What units would you use with your class?

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

9. 5a) What should you graph against p to get a straight line? Hint: It isn’t any power of v.

10. 5d) What is the physical meaning of the slope? The mass of a muon is 105.7 MeV/c2

11. You have just shown that p = gmv. Sometimes this is interpreted as mass increases with speed or p = (gm)v.

12. 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

13. Relativity and Energy: Fermilab“Finding the Top Quark” 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

14. 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.

15. 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.

16. The proton-antiproton pair annihilate and a top-antitop pair appear, moving very slowly. Which way will they move? in the same directions as the original pair opposite the proton-antiproton velocities opposite each other There is not enough information Momentum must be conserved. They must move opposite each other.

17. 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!

18. This is the standard

19. When you add the five momenta you should find that they add to A) around 300 GeV B) around 0 GeV C) exactly 0 GeV D) a different number each time C) exactly 0 GeVconservation of p B) around 0 GeVexperimental error

20. 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.

21. 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.

22. 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.

23. This is the standard

24. 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) 2The equation can be simplified to E = pc.

25. 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.

26. This energy produced an almost stationary pair of top-antitop quarks. What is the mass of the top quark?A) 330 GeV B) 115 GeV C) 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.

27. 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.

28. 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.

29. Relativity and Time: The GPS“Everyday Einstein” 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

30. 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? undefined B) infinite C) zero D) one D) one

31. 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.

32. When v is much less than c, the relationship is well approximated by Calculate gamma. Most calculators give 1.000000000.

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

34. What should you enter into your calculator? You only have 4 digits for the satellite speed. Powers of ten should be treated separately.

35. The correct answer is A) 8.34939395 x 10-11 s B) 8.349 x 10-11 s C) 8.35 x 10-11 s D)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.

36. 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.

37. Frames of Reference“Everyday Einstein” B3.1 distinguish between reference systems (inertial and non-inertial) with respect to real and apparent forces acting within such systems.

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

39. The bottle is thrown upwards and the hole is uncovered. What happens to the water while the bottle is in the air? It pours out on the way up and down. It stays in the bottle. It pours out only on the way up. It pours out only on the way down.

40. 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

41. 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).

42. 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

43. 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.

44. 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.

45. On the left are the positions of a rocket moving up at a constant velocity and shown every 100 ns. 2 3 3 2 1 1 2 3 3 5 2 1 4 4 4 4 5 6 5 5 1 Fig. 1: A rocket moving with constant velocity. Fig. 2: A rocket moving with constant acceleration. 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 nsB) more frequentlyC) less frequently A) You can’t tell if your frame is moving.

46. 2 5 3 3 2 1 1 2 3 3 5 1 1 4 4 4 4 5 6 5 2 Fig. 2: A rocket moving with constant acceleration. Fig. 1: A rocket moving with constant velocity. On the right are the positions of the rocket when it is accelerating upwards. How often does Alice receive the pulses now? every 100 nsB) more frequentlyC) less frequently C) The light must cover extra distance.

47. The rocket is stationary in a gravitational field. How often does Alice receive the pulses? every 100 nsB) more frequentlyC) less frequently Hint: 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.

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

49. 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.

50. Curved Space-Time: “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)