The Matter-Energy Interface Study Questions
Short Answer
1. A cube of aluminum 1.00 m × 1.00 m × 1.00 m is moving at 0.90c, in an orientation as shown. The rest
density of aluminum is 2.70 × 103 kg/m3.
(a) Which of its three dimensions, a, b, or c, is affected by its motion?
(b) Calculate its relativistic volume.
(c) Calculate its relativistic mass.
(d) What is its density at v = 0.90c?
2. A spaceship passes you at a speed of 0.90c, and you measure its length to be 50 m. What is its measured
length when at rest?
3. A beam of unknown elementary particles travels at a speed of 2.0 × 108 m/s. When the particles are moving at
this speed, their average lifetime is found by measurement to be 1.6 × 10–8 s. What is their average lifetime
when at rest?
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
0.950
0.960
0.970
0.980
0.990
0.995
0.010
0.040
0.090
0.160
0.250
0.360
0.490
0.640
0.810
0.902
0.923
0.941
0.960
0.980
0.990
0.990
0.960
0.910
0.840
0.750
0.640
0.510
0.360
0.190
0.098
0.077
0.059
0.040
0.020
0.010
0.995
0.980
0.954
0.917
0.866
0.800
0.714
0.600
0.436
0.31
0.28
0.24
0.20
0.14
0.10
1.01
1.02
1.05
1.09
1.15
1.25
1.40
1.67
2.29
3.2
3.6
4.2
5.0
7.1
10
4. A distant star has been measured as moving at a speed of 0.8c away from Earth. If a planet of this star sends
radio signals to Earth once every second (according to their time scale), how often would we receive them
(according to our time scale)?
5. The rest mass of a proton is 1.67 × 10–27 kg. What would we measure as the mass of a proton moving at
0.90c?
6. Tiny subatomic particles called mu-mesons (or muons) are created in collisions between cosmic rays and
atoms at the upper limit of our atmosphere. At rest, muons have a mean lifetime of only 2.2 × 10–6 s, before
disintegrating into other particles.
(a) What is their mean lifetime, if they travel at 0.99c?
(b) Not taking relativity into account, what is the average distance you would expect muons to move, at this
speed, before disintegrating?
(c) What actual distance, on the average, do they move before breaking apart?
7. A star is 40.0 light-years from Earth (1 light-year is the distance that light travels in 1 year).
(a) How far would you measure this distance to be if you travelled it in a spaceship moving at 1.00 × 108 m/s?
(b) How long would the trip last (for you)?
8. A spaceship goes past a planet at a speed of 0.80c. An observer on the planet measures the length of the
moving spaceship as 40 m. He also says that his planet has a diameter of 2.0 × 106 m.
(a) How long does the woman on the spaceship measure the ship to be?
(b) What does the woman on the spaceship measure the diameter of the planet to be?
(c) According to the man on the planet, the spaceship takes 8.0 s to reach the next planet in his solar system.
How long would the woman on the spaceship say it took?
9. How much energy can be produced by the complete annihilation of 1.0 kg of mass?
10. A nuclear generating plant produces electric power at an average rate of 5.0 GW, by converting the mass of
nuclear fuel. How much fuel is converted to energy in one year, if the process is assumed to be 100%
efficient?
11. An astronaut who was 20 years old left to explore the galaxy in 1980, on a spaceship travelling at 2.5 × 108
m/s. He returns in 2020. About how old will he appear to be?
12. The Earth, of mass 5.98 × 1024 kg, moves along its solar orbit at an average speed of 2.96 × 104 m/s. How
much mass, if converted into energy, could accelerate Earth from rest to that speed?
13. Scientist Ludwig von Drake, while in his laboratory, measures the half-life of some radioactive material
which is in a bomb, approaching with speed v. Donald Duck, who is riding on the bomb, also measures the
half-life. His answer is a factor of two smaller than Ludwig’s. What is the value of v, expressed as a fraction
of c?
14. How much would the energy produced by the complete annihilation of 1.0 kg of mass cost, at 5¢/kW·h, a
typical utility price?
15. Determine the energy, in eV, for photons with the following characteristics.
(a) 1.2 × 1018 Hz (X-rays)
(b) 400 nm (violet)
(c) 4.4 × 1014 Hz (red)
(d) 900 nm (infrared)
16. Calculate the wavelength, in nanometres, of a photon with 3.20 × 10–19 J of energy.
17. What is the minimum frequency of the photon required to eject electrons from a metal whose work function is
2.4 eV?
18. Barium has a work function of 2.48 eV. What is the maximum kinetic energy of the ejected electrons if the
metal is illuminated by light with a wavelength of 450 nm?
19. When 350 nm light falls on a metal, the maximum kinetic energy of the ejected electrons is 1.20 eV. What is
the work function of the metal?
20. What is the momentum of a photon that has a wavelength of 1.2 × 10–12 m?
21. Calculate the momentum of a photon whose wavelength is 500 nm.
22. What is the momentum of a photon with a frequency of 4.5 × 1015 Hz?
23. What is the momentum of a 150 eV photon?
24. Calculate the wavelength of a photon having the same momentum as an electron moving at 1.0 × 106 m/s.
25. What is the de Broglie wavelength of a 0.10 kg ball moving at 20 m/s?
26. Calculate the de Broglie wavelength of each of the following.
(a) a 2.0 kg ball thrown at 15 m/s
(b) a proton accelerated to 1.3 × 105 m/s (mp = 1.7 × 10–27 kg)
(c) an electron moving at 5.0 × 104 m/s
27. Calculate the energy of an ultraviolet photon of light whose wavelength is 122 nm. Express your answer in
both joules and electron volts.
28. What are the energies of the photons emitted by two radio stations with the following frequencies?
(a) 570 kHz
(b) 102 MHz
Express your answers in electron volts.
29. Show that
, where E is the energy of a photon in electron volts, and λ is its wavelength in
nanometres.
30. What is the minimum frequency of the light required to eject photoelectrons from a metallic surface whose
work function is 7.2 × 10–19 J?
31. What is the wavelength of light that ejects photoelectrons from a tungsten surface (W = 4.52 eV), when the
maximum kinetic energy of the electrons is 1.68 eV?
32. When light with a wavelength of 480 nm falls on a metallic surface, a retarding potential of 1.2 V is required
to make the current passing through the phototube zero. What is the work function of the metal?
33. What is the momentum of a 410 nm photon of violet light?
34. (a) What is the momentum of an electron that has an associated de Broglie wavelength of 1.0 × 10–10 m?
(b) What is the velocity of the same electron?
(c) What is the kinetic energy of the same electron?
35. What is the associated de Broglie wavelength of
(a) a neutron (mn = 1.67 × 10–27 kg) travelling at 1.5 × 104 m/s?
(b) an electron (me = 9.11 × 10–31 kg) travelling at 1.2 × 106 m/s?
(c) a proton whose kinetic energy is 1.0 × 109 eV?
(d) an artillery shell with a mass of 0.50 kg and a velocity of 500 m/s?
36. What is the energy, in electron volts, required to give an electron an associated de Broglie wavelength of 0.15
nm?
37. According to the Bohr theory of the atom, the velocity of an electron in the first Bohr orbit of the hydrogen
atom is 2.19 × 106 m/s.
(a) What is the de Broglie wavelength associated with this electron?
(b) The radius of the first Bohr orbit is 5.3 × 10–11 m. How does the de Broglie wavelength of the electron
compare with the circumference of the first orbit?
38. Calculate the energy of
(a) a photon of blue light with a frequency of 6.67 × 1014 Hz
(b) a photon of red light with a wavelength of 630 nm
39. What is the frequency of a photon with a wavelength of 2.0 × 10–7 m?
40. What is the threshold frequency for a calcium surface whose work function is 3.33 eV?
41. Light with a wavelength of 600 nm strikes a metal having a work function of 2.3 × 10–19 J. Calculate the
maximum kinetic energy, in joules, of the emitted electrons and the voltage required to stop them.
42. Light with a wavelength of 430 nm falls on a photoelectric surface. The maximum kinetic energy of the
photoelectrons is 1.21 eV. What is the work function of the surface?
Problem
43. An astronaut whose pulse remains constant at 72 beats/min is sent on a voyage in a spaceship, and his pulse is
measured by a stationary observer on Earth. What would his pulse beat be when the ship is moving, relative
to Earth, at
(a) 0.10c
(b) 0.90c
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
0.950
0.960
0.970
0.980
0.990
0.995
0.010
0.040
0.090
0.160
0.250
0.360
0.490
0.640
0.810
0.902
0.923
0.941
0.960
0.980
0.990
0.990
0.960
0.910
0.840
0.750
0.640
0.510
0.360
0.190
0.098
0.077
0.059
0.040
0.020
0.010
0.995
0.980
0.954
0.917
0.866
0.800
0.714
0.600
0.436
0.31
0.28
0.24
0.20
0.14
0.10
1.01
1.02
1.05
1.09
1.15
1.25
1.40
1.67
2.29
3.2
3.6
4.2
5.0
7.1
10
44. In SI, the standard units for mass, length, and time are the kilogram, metre, and second. In fact, these
standards are defined “at rest.” Using the equations developed in your study of relativity, complete the
following table to show how each of these quantities change as
.
v
L
m
t
0
1m
1 kg
1s
0.10c
0.50c
0.80c
0.90c
0.95c
0.98c
0.99c
1.00c
On the same set of axes, labelled separately for each unit, draw graphs of L, m, and t versus v, from v = 0 to v
= c.
Note: All values of
shall be taken from the table above, where possible.
45. It is anticipated that one day intercontinental jet aircraft may be able to cruise at speeds of 4000 km/h
(1.1 × 103 m/s). By how much would such a jet appear shortened, when viewed from the ground, if its proper
length is 100 m?
46. A K+ meson has an average rest lifetime of 1.0 × 10–8 s. How fast must it be moving to double its average
lifetime?
47. A spaceship must be launched from Earth at a speed of 1.1 × 104 m/s in order to escape from Earth’s
gravitational field. By how much would the mass of a 3.0 × 105 kg spaceship increase when launched at this
speed? (Ignore the mass of fuel burned.)
48. (a) With what speed must an electron be moving in order that its relativistic mass be 102 times its rest mass?
(b) To achieve this speed, electrons may be accelerated through a tube that is 3.0 km long. How long does this
tube seem to be to the electron moving at its final speed?
49. Light with a wavelength of 600 nm is directed at a metallic surface with a work function of 1.60 eV. Calculate
(a) the maximum kinetic energy, in joules, of the emitted electrons
(b) their maximum speed
(c) the cut-off potential necessary to stop these electrons
50. Light of frequency 8.0 × 1014 Hz illuminates a surface whose work function is 1.2 eV. If the retarding
potential is 1.0 V, what is the maximum velocity with which an electron reaches the plate?
51. What is the de Broglie wavelength of an electron that has been accelerated from rest through a potential
difference of 50 V?
52. What are the wavelengths, in metres, of a 3.0 eV photon and a 5.0 eV electron?
53. The average solar power received at ground level in Toronto and Montreal is about 1.0 kW/m2.
(a) If the average wavelength of light is taken as 550 nm, how many photons strike an area of 1.0 cm2 per
second? (Assume that the light rays strike perpendicularly to the surface.)
(b) How many photons would you find in a thimble with a volume of 1 cm3?
54. What is the momentum and the equivalent mass of a 0.20 nm X-ray photon? (This does not imply a mass for
the photon!)
55. An electron is accelerated from rest through a potential difference of 100 V. What is the associated de Broglie
wavelength of the electron?
The Matter-Energy Interface Study Questions
Answer Section
SHORT ANSWER
1. ANS:
(a) Only dimensions in the direction of motion are affected, therefore b.
(b)
(c)
(d)
REF: K/U
MSC: P
2. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.5, p.686
REF: K/U
MSC: P
3. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.691
REF: K/U
MSC: P
4. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.691
REF: K/U
MSC: P
5. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.691
REF: K/U
MSC: P
6. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.691
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.692
(a)
(b)
(c)
REF: K/U, MC
MSC: P
7. ANS:
(a)
(b) For you, at
:
REF: K/U
MSC: P
8. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.692
(a)
Since the stationary observer measures relativistic length of the spaceship.
(b)
Since woman in spaceship would be observing relativistic diameter of planet
(c)
REF: K/U
MSC: P
9. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.692
REF: K/U
MSC: P
10. ANS:
OBJ: 11.3
LOC: ME1.06
KEY: FOP 17.8, p.693
REF: K/U
MSC: P
11. ANS:
OBJ: 11.3
LOC: ME1.06
KEY: FOP 17.8, p.693
Therefore, age upon return = 20a + 22a = 42a
REF: K/U
MSC: P
12. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.692
REF: K/U
MSC: P
13. ANS:
OBJ: 11.3
LOC: ME1.06
KEY: FOP 17.8, p.693
REF: K/U
MSC: P
14. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.693
REF: K/U
MSC: P
OBJ: 11.3
LOC: ME1.06
KEY: FOP 17.8, p.693
15. ANS:
(a)
(b)
(c)
(d)
REF: K/U
MSC: P
16. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.2, p.705
REF: K/U
MSC: P
17. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.2, p.705
REF: K/U
MSC: P
18. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.2, p.705
REF: K/U
MSC: P
19. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.2, p.705
REF: K/U
MSC: P
20. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.2, p.705
REF: K/U
MSC: SP
21. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.3, p.707
REF: K/U
MSC: P
22. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.3, p.707
REF: K/U
MSC: P
23. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.3, p.707
REF: K/U
MSC: P
24. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.3, p.707
REF: K/U
MSC: P
25. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.3, p.707
Note that for everyday objects the wavelength is extremely small.
REF: K/U
MSC: SP
26. ANS:
(a)
(b)
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.5, p.714
(c)
REF: K/U
MSC: SP
27. ANS:
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.5, p.714
REF: K/U
MSC: P
28. ANS:
(a)
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.728
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.728
(b)
REF: K/U
MSC: P
29. ANS:
REF: K/U
MSC: P
30. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.728
REF: K/U
MSC: P
31. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.728
REF: K/U
MSC: P
32. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.729
REF: K/U
MSC: P
33. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.729
REF: K/U
MSC: P
34. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.8, p.729
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.8, p.729
(a)
(b)
(c)
REF: K/U
MSC: P
35. ANS:
(a)
(b)
(c)
(d)
REF: K/U
MSC: P
36. ANS:
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.8, p.729
REF: K/U
MSC: P
37. ANS:
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.8, p.729
(a)
(b)
In other words, the circumference of the first orbit in the hydrogen atom is equal to the de Broglie wavelength
of the electron.
REF: K/U
MSC: P
38. ANS:
(a)
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.8, p.729
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.2, p.703
(b)
REF: K/U
MSC: SP
39. ANS:
REF: K/U
MSC: P
40. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.8, p.728
REF: K/U
MSC: P
41. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.728
Therefore the required potential difference is –0.64 V. (The negative potential is required to stop the
electrons.)
REF: K/U
MSC: P
42. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.729
REF: K/U
MSC: P
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.729
PROBLEM
43. ANS:
The time between pulse beats, measured on the spaceship, is:
Then, at v = 0.10c:
Thus, as measured from Earth, his pulse rate, at v = 0.10c, remains at 72 beats/min.
At v = 0.90c:
Therefore, as measured from Earth, at v = 0.90c, his pulse rate is 31 beats/min, much slower than the 72
beats/min that the astronaut would measure for himself on the speeding spaceship.
Note that time dilation effects only become noticeable at speeds closely approaching the speed of light, and
do not, therefore, form any part of our everyday experience.
REF: K/U
MSC: SP
44. ANS:
For length:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.5, p.682
For mass:
For time:
Using the same equations for the other values of v,
v
L
m
t
0.10c
0.995 m
1.01 kg
1.01 s
0.50c
0.866 m
1.15 kg
1.15 s
0.80c
0.600 m
1.67 kg
1.67 s
0.90c
0.436 m
2.29 kg
2.29 s
0.95c
0.31 m
3.2 kg
3.2 s
0.98c
0.20 m
5.0 kg
5.0 s
0.99c
0.14 m
7.1 kg
7.1 s
1.00c
0
REF: K/U, C
MSC: P
45. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.5, p.685
Note: In problems such as this where v << c, the binomial expansion of the
term may be used.
We can show that:
REF: K/U, MC
MSC: P
46. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.691
REF: K/U
MSC: P
47. ANS:
OBJ: 11.2
But, using the binomial expansion:
LOC: ME1.05
KEY: FOP 17.8, p.692
REF: K/U
MSC: P
48. ANS:
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.692
OBJ: 11.2
LOC: ME1.05
KEY: FOP 17.8, p.692
(a)
REF: K/U
MSC: P
49. ANS:
(a)
(b)
(c)
Note that the answer to (c) could have been determined from (a), as follows: 7.6 × 10–20 J = 0.48 eV. Thus the
cut-off potential for electrons would be 0.48 V.
REF: K/U
MSC: SP
50. ANS:
OBJ: 12.1
LOC: ME1.03
Maximum energy of photoelectrons which reach the plate will be:
KEY: FOP 18.2, p.704
REF: K/U
OBJ: 12.1
MSC: P
51. ANS:
The energy of the electron is:
LOC: ME1.03
KEY: FOP 18.2, p.705
Note that this wavelength is appreciable on the scale of atoms.
REF: K/U
MSC: SP
52. ANS:
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.5, p.714
REF: K/U
MSC: P
53. ANS:
(a)
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.5, p.714
photons in a box whose dimensions are 1 cm × 1 cm × 3.00 ×
(b) There are
number of photons in a thimble whose dimensions are 1 cm × 1 cm × 1 cm is
REF: K/U
MSC: P
54. ANS:
OBJ: 12.1
LOC: ME1.03
KEY: FOP 18.8, p.728
REF: K/U
MSC: P
55. ANS:
OBJ: 12.1
LOC: ME1.01
KEY: FOP 18.8, p.729
. Thus, the
REF: K/U
MSC: P
OBJ: 12.2
LOC: ME1.01
KEY: FOP 18.8, p.729