Problems & Exercises

Problems & Exercises

11.1 Einstein’s Postulates

1

Under what conditions should experimental results and theoretical predictions be looked at through the lens of special relativity rather than classical mechanics? What breaks down under classical mechanics that can be addressed with special relativity?

11.2 Simultaneity and Time Dilation

2

(a) What is γγ size 12{γ} {} if v=0.250cv=0.250c size 12{v=0 "." "250"c} {}? (b) If v=0.500cv=0.500c size 12{v=0 "." "500"c} {}?

3

(a) What is γγ size 12{γ} {} if v=0.100cv=0.100c size 12{v=0 "." "100"c} {}? (b) If v=0.900cv=0.900c size 12{v=0 "." "900"c} {}?

4

Particles called ππ size 12{π} {}-mesons are produced by accelerator beams. If these particles travel at 2.70×108m/s2.70×108m/s size 12{2 "." "70" times "10" rSup { size 8{8} } `"m/s"} {} and live 2.60×108s2.60×108s when at rest relative to an observer, how long do they live as viewed in the laboratory?

5

Suppose a particle called a kaon is created by cosmic radiation striking the atmosphere. It moves by you at 0.980c,0.980c, size 12{0 "." "980"c} {} and it lives 1.24×108s1.24×108s when at rest relative to an observer. How long does it live as you observe it?

6

A neutral ππ size 12{π} {}-meson is a particle that can be created by accelerator beams. If one such particle lives 1.40×1016s1.40×1016s as measured in the laboratory, and 0.840×1016s0.840×1016s when at rest relative to an observer, what is its velocity relative to the laboratory?

7

A neutron lives 900 s when at rest relative to an observer. How fast is the neutron moving relative to an observer who measures its life span to be 2,065 s?

8

If relativistic effects are to be less than 1 percent, then γγ size 12{γ} {} must be less than 1.01 percent. At what relative velocity is γ=1.01γ=1.01 size 12{γ=1 "." "01"} {}?

9

If relativistic effects are to be less than 3 percent, then γγ size 12{γ} {} must be less than 1.03 percent. At what relative velocity is γ=1.03γ=1.03 size 12{γ=1 "." "03"} {}?

10

(a) At what relative velocity is γ=1.50γ=1.50 size 12{γ=1 "." "50"} {}? (b) At what relative velocity is γ=100γ=100 size 12{γ="100"} {}?

11

(a) At what relative velocity is γ=2.00γ=2.00 size 12{γ=2 "." "00"} {}? (b) At what relative velocity is γ=10.0γ=10.0 size 12{γ="10" "." 0} {}?

12

Unreasonable Results

(a) Find the value of γγ size 12{γ} {} for the following situation. An Earth-bound observer measures 23.9 h to have passed while signals from a high-velocity space probe indicate that 24.0 h24.0 h size 12{"24" "." 0" h"} {} have passed on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

11.3 Length Contraction

13

A spaceship, 200 m long as seen on board, moves by the Earth at 0.970c.0.970c. size 12{0 "." "970"c} {} What is its length as measured by an Earth-bound observer?

14

How fast would a 6.0 m-long sports car have to be going past you in order for it to appear only 5.5 m long?

15

(a) How far does the muon in Example 11.1 travel according to the Earth-bound observer? (b) How far does it travel as viewed by an observer moving with it? Base your calculation on its velocity relative to Earth and the time it lives (proper time). (c) Verify that these two distances are related through length contraction γ = 3.20γ = 3.20 size 12{ ital "γ="3 "." "20"} {}.

16

(a) How long would the muon in Example 11.1 have lived as observed on Earth if its velocity was 0.0500c0.0500c size 12{0 "." "0500"c} {}? (b) How far would it have traveled as observed on Earth? (c) What distance is this in the muon’s frame?

17

(a) How long does it take the astronaut in Example 11.2 to travel 4.30 ly at 0.99944c0.99944c (as measured by Earth-bound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with γ = 30.00γ = 30.00 size 12{ ital "γ=""30" "." "00"} {} as given.

18

(a) How fast would an athlete need to be running for a 100-m race to look 100 yd long? (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary circumstances? Explain.

19

Unreasonable Results

(a) Find the value of γγ size 12{γ} {} for the following situation. An astronaut measures the length of her spaceship to be 25 m, while an Earth-bound observer measures it to be 100 m. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

20

Unreasonable Results

A spaceship is heading directly toward Earth at a velocity of 0.800c.0.800c. size 12{0 "." "800"c} {} The astronaut on board claims that he can send a canister toward Earth at 1.20c1.20c size 12{1 "." "20"c} {} relative to Earth. (a) Calculate the velocity the canister must have relative to the spaceship. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

11.4 Relativistic Addition of Velocities

21

Suppose a spaceship heading straight toward Earth at 0.750c0.750c size 12{0 "." "750"c} {} can shoot a canister at 0.500c0.500c size 12{0 "." "750"c} {} relative to the ship. (a) What is the velocity of the canister relative to Earth, if it is shot directly at Earth? (b) If it is shot directly away from Earth?

22

Repeat the previous problem with the ship heading directly away from Earth.

23

If a spaceship is approaching the Earth at 0.100c0.100c and a message capsule is sent toward it at 0.100c0.100c relative to the Earth, what is the speed of the capsule relative to the ship?

24

(a) Suppose the speed of light were only 3,000 m/s3,000 m/s. A jet fighter moving toward a target on the ground at 800 m/s800 m/s shoots bullets, each having a muzzle velocity of 1,000 m/s1,000 m/s. What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.

25

If a galaxy moving away from Earth has a speed of 1,000 km/s1,000 km/s size 12{1,"000"" m/s"} {} and emits 656 nm656 nm size 12{"656"" nm"} {} light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of Earth in its orbit negligible here?

26

A space probe speeding towards the nearest star moves at 0.250c0.250c size 12{0 "." "250"c} {} and sends radio information at a broadcast frequency of 1 GHz. What frequency is received on the Earth?

27

If two spaceships are heading directly toward each other at 0.800c,0.800c, size 12{0 "." "800"c} {} at what speed must a canister be shot from the first ship to approach the other at 0.999c0.999c size 12{0 "." "999"c} {} as seen by the second ship?

28

Two planets are on a collision course, heading directly towards each other at 0.250c.0.250c. size 12{0 "." "250"c} {} A spaceship sent from one planet approaches the second at 0.750c0.750c size 12{0 "." "750"c} {} as seen by the second planet. What is the velocity of the ship relative to the first planet?

29

When a missile is shot from one spaceship towards another, it leaves the first at 0.950c0.950c size 12{0 "." "950"c} {} and approaches the other at 0.750c0.750c size 12{0 "." "750"c} {}. What is the relative velocity of the two ships?

30

Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1,900-nm electromagnetic radiation and know that it was 1,875-nm when emitted by the hydrogen gas. What is the speed of the gas?

31

A highway patrol officer uses a device that measures the speed of vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar has a frequency of 100 GHz and the returning echo has a frequency 15 kHz higher. What is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain not to round off until the end of the problem, because the effect is small.

32

Prove that for any relative velocity vv size 12{v} {} between two observers, a beam of light sent from one to the other will approach at speed c,c, size 12{c} {} provided that vv size 12{v} {} is less than c.c. size 12{c} {}

33

Show that for any relative velocity vv size 12{v} {} between two observers, a beam of light projected by one directly away from the other will move away at the speed of light, provided that vv size 12{v} {} is less than c.c. size 12{c} {}

34

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy 12.0×109 12.0×109 size 12{"12" "." 0 times "10" rSup { size 8{9} } " ly"} {} ly away is receding from us at .900c,.900c, at what velocity relative to us must we send an exploratory probe to approach the other galaxy at .990c,.990c, as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? All of this is possible in principle, but not practical.

11.5 Relativistic Momentum

35

Find the momentum of a helium nucleus having a mass of 6.68×10–27 kg6.68×10–27 kg size 12{6 "." "68" times "10" rSup { size 8{"27"} } " kg"} {} that is moving at 0.200c.0.200c. size 12{0 "." "200"c} {}

36

What is the momentum of an electron traveling at 0.980c.0.980c. size 12{0 "." "980"c} {}?

37

(a) Find the momentum of a 1.00×109 kg1.00×109 kg size 12{1 "." "00" times "10" rSup { size 8{9} } " kg"} {} asteroid heading toward Earth at 30 km/s30 km/s size 12{"30" "." 0" km/s"} {}. (b) Find the ratio of this momentum to the classical momentum. Hint: Use the approximation that γ=1+(1/2)v2/c2γ=1+(1/2)v2/c2 size 12{γ=1+ \( 1/2 \) v rSup { size 8{2} } /c rSup { size 8{2} } } {} at low velocities.

38

(a) What is the momentum of a 2000-kg satellite orbiting at 4 km/s? (b) Find the ratio of this momentum to the classical momentum. (Hint—Use the approximation that γ=1+(1/2)v2/c2γ=1+(1/2)v2/c2 size 12{γ=1+ \( 1/2 \) v rSup { size 8{2} } /c rSup { size 8{2} } } {} at low velocities.)

39

What is the velocity of an electron that has a momentum of 3.04×10–21 kg⋅m/s3.04×10–21 kg⋅m/s size 12{3 "." "34" times "10" rSup { size 8{"21"} } " kg" "." "m/s?"} {}? Note that you must calculate the velocity to at least four digits to see the difference from cc.

40

Find the velocity of a proton that has a momentum of 4.48×–10-19 kg⋅m/s.4.48×–10-19 kg⋅m/s. size 12{4 "." "48" times "10" rSup { size 8{"19"} } " kg" "." "m/s" "." } {}

41

(a) Calculate the speed of a 1.00-μg1.00-μg particle of dust that has the same momentum as a proton moving at 0.999c0.999c size 12{0 "." "999"c} {}. (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

42

(a) Calculate γγ size 12{γ} {} for a proton that has a momentum of 1.00 kg⋅m/s.1.00 kg⋅m/s. size 12{1 "." "00 kg" "." "m/s" "." } {} (b) What is its speed? Such protons form a rare component of cosmic radiation with uncertain origins.

11.6 Relativistic Energy

43

What is the rest energy of an electron, given its mass is 9.11×1031 kg9.11×1031 kg? Give your answer in joules and MeV.

44

Find the rest energy in joules and MeV of a proton, given its mass is 1.67×1027 kg1.67×1027 kg.

45

If the rest energies of a proton and a neutron, the two constituents of nuclei, are 938.3 and 939.6 MeV respectively, what is the difference in their masses in kilograms?

46

The Big Bang that began the universe is estimated to have released 1068 J1068 J size 12{10 rSup { size 8{"68"} } " J"} {} of energy. How many stars could half of this energy create, assuming the average star’s mass is 4.00×1030 kg4.00×1030 kg size 12{4 "." "00" times "10" rSup { size 8{"30"} } " kg"} {}?

47

A supernova explosion of a 2.00×1031 kg2.00×1031 kg size 12{2 "." "00" times "10" rSup { size 8{"31"} } " kg"} {} star produces 1.00×1044 kg1.00×1044 kg size 12{1 "." "00" times "10" rSup { size 8{"44"} } " kg"} {} of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio Δm/mΔm/m size 12{Δm/m} {} of mass destroyed to the original mass of the star?

Object/Phenomenon Energy in Joules
Big Bang 10 68 10 68 size 12{"10" rSup { size 8{"68"} } } {}
Energy released in a supernova 10 44 10 44 size 12{"10" rSup { size 8{"44"} } } {}
Fusion of all the hydrogen in Earth’s oceans 10 34 10 34 size 12{"10" rSup { size 8{"34"} } } {}
Annual world energy use 4 × 10 20 4 × 10 20 size 12{4 times "10" rSup { size 8{"20"} } } {}
Large fusion bomb (9 megaton) 3 . 8 × 10 16 3 . 8 × 10 16 size 12{3 "." 8 times "10" rSup { size 8{"16"} } } {}
1 kg hydrogen (fusion to helium) 6 . 4 × 10 14 6 . 4 × 10 14 size 12{6 "." 4 times "10" rSup { size 8{"14"} } } {}
1 kg uranium (nuclear fission) 8 . 0 × 10 13 8 . 0 × 10 13 size 12{8 "." 0 times "10" rSup { size 8{"13"} } } {}
Hiroshima-size fission bomb (10 kiloton) 4 . 2 × 10 13 4 . 2 × 10 13 size 12{4 "." 2 times "10" rSup { size 8{"13"} } } {}
90,000-ton aircraft carrier at 30 knots 1 . 1 × 10 10 1 . 1 × 10 10 size 12{1 "." 1 times "10" rSup { size 8{"10"} } } {}
1 barrel crude oil 5 . 9 × 10 9 5 . 9 × 10 9 size 12{5 "." 9 times "10" rSup { size 8{9} } } {}
1 ton TNT 4 . 2 × 10 9 4 . 2 × 10 9 size 12{4 "." 2 times "10" rSup { size 8{9} } } {}
1 gallon of gasoline 1 . 2 × 10 8 1 . 2 × 10 8 size 12{1 "." 2 times "10" rSup { size 8{8} } } {}
Daily home electricity use (developed countries) 7 × 10 7 7 × 10 7 size 12{7 times "10" rSup { size 8{7} } } {}
Daily adult food intake (recommended) 1 . 2 × 10 7 1 . 2 × 10 7 size 12{1 "." 2 times "10" rSup { size 8{7} } } {}
1000-kg car at 90 km/h 3 . 1 × 10 5 3 . 1 × 10 5 size 12{3 "." 1 times "10" rSup { size 8{5} } } {}
1 g fat (9.3 kcal) 3 . 9 × 10 4 3 . 9 × 10 4 size 12{3 "." 9 times "10" rSup { size 8{4} } } {}
ATP hydrolysis reaction 3 . 2 × 10 4 3 . 2 × 10 4 size 12{3 "." 2 times "10" rSup { size 8{4} } } {}
1 g carbohydrate (4.1 kcal) 1 . 7 × 10 4 1 . 7 × 10 4 size 12{1 "." 7 times "10" rSup { size 8{4} } } {}
1 g protein (4.1 kcal) 1 . 7 × 10 4 1 . 7 × 10 4 size 12{1 "." 7 times "10" rSup { size 8{4} } } {}
Tennis ball at 100 km/h 22 22
Mosquito (10–2g at 0.5 m/s)(10–2g at 0.5 m/s) 1 . 3 × 10 6 1 . 3 × 10 6 size 12{1 "." 3 times "10" rSup { size 8{-6} } } {}
Single electron in a TV tube beam 4 . 0 × 10 15 4 . 0 × 10 15 size 12{4 "." 0 times "10" rSup { size 8{-"15"} } } {}
Energy to break one DNA strand 10 19 10 19 size 12{"10" rSup { size 8{-"19"} } } {}
Table 11.1 Energy of Various Objects and Phenomena
48

(a) Using data from Table 11.1, calculate the mass converted to energy by the fission of 1 kg of uranium. (b) What is the ratio of mass destroyed to the original mass, Δm/mΔm/m?

49

(a) Using data from Table 11.1, calculate the amount of mass converted to energy by the fusion of 1 kg of hydrogen. (b) What is the ratio of mass destroyed to the original mass, Δm/mΔm/m? (c) How does this compare with Δm/mΔm/m for the fission of 1 kg of uranium?

50

There is approximately 1034 J1034 J of energy available from fusion of hydrogen in the world’s oceans. (a) If 1033 J1033 J size 12{10 rSup { size 8{"33"} } " J"} {} of this energy were utilized, what would be the decrease in mass of the oceans? Assume that 0.08 percent of the mass of a water molecule is converted to energy during the fusion of hydrogen. (b) How great a volume of water does this correspond to? (c) Comment on whether this is a significant fraction of the total mass of the oceans.

51

A muon has a rest mass energy of 105.7 MeV, and it decays into an electron and a massless particle. (a) If all the lost mass is converted into the electron’s kinetic energy, find γγ size 12{γ} {} for the electron. (b) What is the electron’s velocity?

52

A ππ size 12{π} {}-meson is a particle that decays into a muon and a massless particle. The ππ size 12{π} {}-meson has a rest mass energy of 139.6 MeV, and the muon has a rest mass energy of 105.7 MeV. Suppose the ππ size 12{π} {}-meson is at rest and all of the missing mass goes into the muon’s kinetic energy. How fast will the muon move?

53

(a) Calculate the relativistic kinetic energy of a 1,000-kg car moving at 30.0 m/s if the speed of light were only 45.0 m/s. (b) Find the ratio of the relativistic kinetic energy to classical.

54

Alpha decay is nuclear decay in which a helium nucleus is emitted. If the helium nucleus has a mass of 6.80×1027 kg6.80×1027 kg size 12{6 "." "80" times "10" rSup { size 8{ - "27"} } " kg"} {} and is given 5.00 MeV of kinetic energy, what is its velocity?

55

(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given 0.750 MeV of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with the kinetic energy as it compares to the rest mass energy of the electron.

56

A positron is an antimatter version of the electron, having exactly the same mass. When a positron and an electron meet, they annihilate, converting all of their mass into energy. (a) Find the energy released, assuming negligible kinetic energy before the annihilation. (b) If this energy is given to a proton in the form of kinetic energy, what is its velocity? (c) If this energy is given to another electron in the form of kinetic energy, what is its velocity?

57

What is the kinetic energy in MeV of a ππ size 12{π} {}-meson that lives 1.40×1016 s1.40×1016 s size 12{1 "." "40" times 10 rSup { size 8{ - "16"} } " s"} {} as measured in the laboratory, and 0.840×1016 s0.840×1016 s size 12{0 "." "840" times 10 rSup { size 8{ - "16"} } " s"} {} when at rest relative to an observer, given that its rest energy is 135 MeV?

58

Find the kinetic energy in MeV of a neutron with a measured life span of 2,065 s, given its rest energy is 939.6 MeV, and rest life span is 900 s.

59

(a) Show that (pc)2/(mc2)2=γ21(pc)2/(mc2)2=γ21. This means that at large velocities pc>>mc2pc>>mc2. (b) Is EpcEpc size 12{E approx ital "pc"} {} when γ=30.0γ=30.0 size 12{γ="30" "." 0} {}, as for the astronaut discussed in the twin paradox?

60

One cosmic ray neutron has a velocity of 0.250c0.250c relative to Earth. (a) What is the neutron’s total energy in MeV? (b) Find its momentum. (c) Is EpcEpc size 12{E approx ital "pc"} {} in this situation? Discuss in terms of the equation given in part (a) of the previous problem.

61

What is γγ size 12{γ} {} for a proton having a mass energy of 938.3 MeV accelerated through an effective potential of 1.0 TV (teravolt) at Fermilab outside Chicago?

62

(a) What is the effective accelerating potential for electrons at the Stanford Linear Accelerator, if γ=1.00×105γ=1.00×105 size 12{γ=1 "." "00" times "10" rSup { size 8{5} } } {} for them? (b) What is their total energy, which isnearly the same as kinetic in this case, in GeV?

63

(a) Using data from Table 11.1, find the mass destroyed when the energy in a barrel of crude oil is released. (b) Given these barrels contain 200 liters and assuming the density of crude oil is 750 kg/m3750 kg/m3, what is the ratio of mass destroyed to original mass, Δm/mΔm/m size 12{Δm/m} {}?

64

(a) Calculate the energy released by the destruction of 1 kg of mass. (b) How many kilograms could be lifted to a 10 km height by this amount of energy?

65

A Van de Graaff accelerator utilizes a 50 MV potential difference to accelerate charged particles such as protons. (a) What is the velocity of a proton accelerated by such a potential? (b) An electron?

66

Suppose you use an average of 500 kW · h500 kW · h of electric energy per month in your home. (a) How long would 1 g of mass converted to electric energy with an efficiency of 38 percent last you? (b) How many homes could be supplied at the 500 kW · h500 kW · h per month rate for one year by the energy from the described mass conversion?

67

(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35 percent. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power? (b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is 104kg?104kg?

68

Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket’s mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? Assume an orbital altitude of 250 km, and calculate both the kinetic, or classical, and the gravitational potential energy needed. (b) If the ship has a mass of 1.00×105kg1.00×105kg, or 100 tons, what total yield nuclear explosion in tons of TNT is needed?

69

The Sun produces energy at a rate of 4.00×10264.00×1026 size 12{4 "." "00" times "10" rSup { size 8{"26"} } } {} W by the fusion of hydrogen. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the sun is 90 percent hydrogen and half of this can undergo fusion before the sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the sun losing per second? (d) What fraction of its mass will it have lost in the time found in part (b)?

70

Unreasonable Results

A proton has a mass of 1.67×1027kg1.67×1027kg. A physicist measures the proton’s total energy to be 50 MeV. (a) What is the proton’s kinetic energy? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

71

Construct Your Own Problem

Consider a highly relativistic particle. Discuss what is meant by the term highly relativistic. (Note that, in part, it means that the particle cannot be massless.) Construct a problem in which you calculate the wavelength of such a particle and show that it is very nearly the same as the wavelength of a massless particle, such as a photon, with the same energy. Among the things to be considered are the rest energy of the particle (it should be a known particle) and its total energy, which should be large compared to its rest energy.

72

Construct Your Own Problem

Consider an astronaut traveling to another star at a relativistic velocity. Construct a problem in which you calculate the time for the trip as observed on Earth and as observed by the astronaut. Also calculate the amount of mass that must be converted to energy to get the astronaut and ship to the velocity traveled. Among the things to be considered are the distance to the star, the velocity, and the mass of the astronaut and ship. Unless your instructor directs you otherwise, do not include any energy given to other masses, such as rocket propellants.

11.6 Relativistic Energy

73

Recall that the Theory of Conservation of Mass states “matter cannot be created nor destroyed.” Consider the equations E0 = mc2 and E = γmc2, then use them to justify the replacement of the Theory of Conservation of Mass with the Theory of Conservation of Mass-Energy.