Physics 103, Prof. Haroyan
Practivce Final Exam
1. Two events occur in an inertial system K as follows:
Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0
Event 2: x2 = 2a, t2 = 3a/2c, y2 = 0, z2 = 0
Is there a frame K ′ in which the two events occur at the same place? Explain.
2. The total energy of a body is found to be twice its rest energy. How fast is it
moving with respect to the observer?
3. If you were to travel to a star 135 light-years from Earth at a speed of 2.80 ×
108 m/s, what would you measure this distance to be? Ans: 48.5 ly
4. A certain star is 18.6 light-years away. How long would it take a spacecraft
traveling 0.950c to reach that star from Earth, as measured by observers: (a)
on Earth, (b) on the spacecraft? (c) What is the distance traveled according to
observers on the spacecraft? (d) What will the spacecraft occupants compute
their speed to be from the results of (b) and (c)? Ans: 19.6yr, 6.11yr, 5.81 ly,
0.950c
5. A particle of mass m travels at a speed v = 0.26c. At what speed will its
momentum be doubled? Ans: 0.47c
6. What is the momentum of a 950-MeV proton (that is, its kinetic energy is 950
MeV)? Ans: 1.6 GeV/c
7. What is the speed of an electron whose kinetic energy is 1.25 MeV? Ans: 0.957c
8. Barium has a work function of 2.48eV . What is the maximum kinetic energy of
electrons if the metal is illuminated by UV light of wavelength 365nm? What is
their speed? Ans: 0.92eV, 5.7 × 10m/s
9. In the Compton effect, determine the ratio (AA/A) of the maximum change AA
in a photon’s wavelength to the photon’s initial wavelength A, if the photon is
(a) a visiblelight photon with A = 550 nm, (b) an X-ray photon with A = 0.10
nm. Ans: 8.8 × 10−6 , 0.049
10. What is the minimum photon energy needed to produce a µ+ −µ− pair? The mass
of each µ (muon) is 207 times the mass of an electron. What is the wavelength
of such a photon? Ans: 5.86 × 10−15
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11. (a) What is the wavelength of a neutron (m = 1.67 × 10−27 kg) traveling at
8.5 × 104 m/s? (b) The speed of an electron (m = 9.11 × 10−31 kg) in a particle
accelerator is 0.98c. Find its de Broglie wavelength. (Use relativistic momentum.) Ans: 4.7 × 10−12 m, 4.9 × 10−13 m
12. (a) Determine the wavelength of the second Balmer line (n = 4 to n = 2 transition). (b) Calculate the ionization energy of doubly ionized lithium, Li2+, which
has Z = 3. (To ionize the atom means removing the electron, or raising it to
zero energy) Ans: 490nm, 122eV
13. A hydrogen atom has an angular momentum of 5.273 × 10−34 kgm2 /s. According
to the Bohr model, what is the energy (eV) associated with this state? Ans:
0.544eV
14. The neutrons in a parallel beam, each having kinetic energy 0.030 eV, are directed
through two slits 0.60 mm apart. How far apart will the interference peaks be
on a screen 1.0 m away? [Hint: First find the wavelength of the neutron.] Ans:
2.8 × 10−7m
15. (a) A radioactive element undergoes an alpha decay with a lifetime of 12µs.
If alpha particles are emitted with 5.5keV kinetic energy, find the uncertainty
∆E/E in the particle energy.
(b) A free neutron (m = 1.67 × 10−27 kg) has a mean life of 900s. What is the
uncertainty in its mass (in kg)?
(c)An electron and a 140g baseball are each traveling 95m/s measured to a precision of 0.085%. Calculate and compare the uncertainty in position of each.
Ans: 1.0 × 10−14m, 3.00 × 10−10 eV /c2 , 1.4 × 10−3 m, 9.3 × 10−33 m
16. (I) A free electron has a wave function ψ(x) = Asin(2.0×1010 x), where x is given
in meters. Determine the electron’s (a) wavelength, (b) momentum, (c) speed,
and (d) kinetic energy. Ans: 3.1 × 10−10 m, 2.1 × 10−24 kgm/s, 2.3 × 106 m/s,
15eV
17. Write the wave function for (a) a free electron and (b) a free proton, each having
a constant velocity v = 3.0 × 105 m/s.
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Ans: ψ(x) = Asin[2.6 × 109 m−1 )x] + Bcos[(2.6 × 109 m−1 )x], ψ(x) = Asin[(4.7 ×
1012 m−1 )x] + Bcos[(4.7 × 1012 m−1 )x]
18. An electron is trapped in a 1.00nm wide rigid box. Determine the probability
of finding the electron within 0.15nm of the center of the box (on either side of
center) for (a) n = 1, (b) n = 5, and (c) n = 20. (d) Compare to the classical
prediction.
Ans:0.56, 0.24, 0.30, the probabilities approach the classical value for large n
19. Consider a particle that can exist anywhere in space with a wave function given
by
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ψ(x) = b−1/2 |x/b|1/2 e−(x/b) /2
, where b = 1.0nm. (a) Check that the wave function is normalized. (b) What
is the most probable position for the particle in the region x > 0? (c) What is
the probability of finding the particle between x = 0nm and x = 0.50nm?
Ans: 0.71nm, 0.11
20. List the quantum numbers for each electron in the ground state of (a) carbon
(Z = 6), (b) aluminum (Z = 13). Ans:
21. (a) What is the approximate radius of a 112
48 Cd-nucleus? (b) Approximately what
is the value of A for a nucleus whose radius is 3.7 X 10-15 m? Ans: 5.8f m, 29
22. Calculate the total binding energy, and the binding energy per nucleon, for (a)
7
197
3 Li, (b) 79 AU .
Ans: 186.6M eV , 186.6M eV /23, 193.5M eV, 193.5M eV /24
23. What fraction of a sample is left after exactly 6 half-lives? ans: 0.015625
problems on particles will on the final but is not included here.
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