Physics 231 Fall 2018
Midterm 1
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1. Below is an object placed before a diverging lens. Each square box is one unit.
(a) (10 points) Fill in blanks above, paying close attention to signs.
Solution:
f = −5,
s = 15,
h = 6,
s0 = −3.75,
h0 = 1.5,
M = 0.25
(b) (5 points) Draw three principal rays on the diagram to determine the location of
the image graphically.
Solution:
October 5th, 2018
Midterm 1
Name:
2. In a double slit experiment, the slits are one sixth as wide as the separation distance of
their centers.
(a) (5 points) Calculate the number of interference maxima that are observed within
the central diffraction maximum. (Hint: drawing the fringe pattern might help.)
Solution: Since the first diffraction minimum (md = 1) overlaps with some
interference maximum (mi ) right at the edge of the central diffraction maximum,
we can take the ratio
d sin θ
mi λ
=
,
a sin θ
md λ
and since a/d = 1/6, we have mi = 6. That means there are five interference
maxima on either side of the central interference maximum before the first
diffraction minimum, giving a total of 11 maxima.
(a)
11 Maxima
(b) (5 points) Find the number of interference maxima in the first diffraction maximum
adjacent to the central diffraction maximum.
Solution: We expect 5 maxima between mi = 6 and mi = 12, where the next
diffraction minimum (md = 2) occurs.
(b)
5 Maxima
(c) (5 points) Find the ratio of the intensity of the fourth interference maximum from
the central maximum to that of the central maximum.
Solution: At the fourth interference maximum, we have α =
d sin θ = 4λ, so α = 8π. Also,
β=
2π
d sin θ
λ
with
2π
2π 4λ
a
4π
a sin θ =
a
= 8π =
.
λ
λ d
d
3
The intensity of the central maximum is I0 , so the ratio we want is
2
2
I4
sin(2π/3)
27
2 α sin β/2
2
= cos
= cos (4π)
=
= 0.171
I0
2
β/2
2π/3
16π 2
(c)
Physics 231 Fall 2018
October 5th, 2018
I4 /I0 = 0.171
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Midterm 1
Name:
3. In a laboratory frame, a particle moves at 0.9c and lives for 0.5 µs from its creation to
its decay.
(a) (5 points) How far does the particle move in the lab frame?
Solution: ∆x = v∆t = (0.9 · 3 × 108 )(5 × 10−7 ) = 135 m.
(a)
∆x = 135 m
(b) (5 points) In its rest frame, how long did the particle survive?
Solution: Lifetime of the particle in its own frame is the proper time, so ∆t0 =
1
1
∆t, with γ = √1−0.9
2 = 2.294. Hence
γ
∆t0 =
1
(5 × 10−7 ) = 0.218 µs.
2.294
(b)
∆t0 = 0.218 µs
(c) (5 points) In another frame, the particle’s speed was found to be 0.99c. How long
did the particle live in that frame?
Solution: γ2 =
√
1
1−0.992
= 7.089. Hence
∆t2 = γ2 ∆t0 = (7.089)(2.18 × 10−7 ) = 1.545 µs.
(c)
Physics 231 Fall 2018
October 5th, 2018
∆t2 = 1.545 µs
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Midterm 1
Name:
4. An object of mass m1 = 3m0 moves to the right through a laboratory at 0.8c and collides
with an object of mass m2 = 4m0 moving to the left through the laboratory at 0.6c.
Afterward, there are two objects, one of which is a stationary particle of mass m3 = 6m0 .
(a) (5 points) What is the velocity of the other object (don’t forget to indicate direction)?
Solution: Conservation of energy:
γ1 m1 + γ2 m2 = γ3 m3 + γ4 m4
5
5
(3m0 ) + (4m0 ) = 1(6m0 ) + γ4 m4
3
4
4m0 = γ4 m4
Conservation of momentum:
γ1 m1 v1 + γ2 m2 v2 = γ3 m3 v3 + γ4 m4 v4
4
5
3
5
(3m0 )
c + (4m0 ) − c = γ4 m4 v4
3
5
4
5
m0 c = γ4 m4 v4
Divide:
m0 c
γ4 m4 v4
=
γ4 m4
4m0
⇒
v4 =
c
4
⇒
4
γ4 = √
15
Note the positive means motion to the right.
(a)
v4 = c/4
(b) (5 points) What is the mass of the other object?
Solution:
m4 =
√
15m0 ≈ 3.87m0 .
(b)
m4 =
√
15m0
(c) (5 points) Determine the change in kinetic energy in this collision. Be sure to
indicate whether kinetic energy is lost or gained.
Solution: Note ∆K = −∆mc2 , so
√
√
∆K = (3m0 + 4m0 )c2 − (6m0 + 15m0 )c2 = (1 − 15)m0 c2 ≈ −2.87m0 c2 ,
so kinetic energy is lost.
Physics 231 Fall 2018
October 5th, 2018
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Midterm 1
Name:
(c)
Physics 231 Fall 2018
October 5th, 2018
2.87m0 c2 is lost
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Midterm 1
Name:
5. (5 points) Parallel light rays cross interfaces from medium 1 into medium 2 and then
into medium 3, as shown below. What can we say about the relative sizes of the indices
of refraction of these media?
A. n1 > n2 > n3
B. n3 > n2 > n1
C. n2 > n3 > n1
D. n1 > n3 > n2
E. None of the above.
6. (5 points) An object is placed before a thin lens which forms a real image on a screen.
If the bottom half of the lens is covered,
A. the bottom half of the image disappears.
B. the top half of the image disappears.
C. the entire image disappears
D. the brightness of the image decreases.
E. the image becomes blurred.
7. (5 points) An object 4.0 cm in height is placed 8.0 cm in front of a concave spherical
mirror with a focal length of 10.0 cm. What is the position of its image in relation to
the mirror, and what are the characteristics of the image?
A. 10.0 cm on the same side of mirror, real, 6.0 times bigger
B. 10.0 cm on the other side of mirror, virtual, 10.0 times bigger
C. 18.0 cm on the same side of mirror, virtual, 2.25 times bigger
D. 40.0 cm on the other side of mirror, virtual, 5.0 times bigger
E. 40.0 cm on the other side of mirror, real, 6.0 times bigger
8. (5 points) A converging lens always produces
A. a magnified image.
B. an image smaller than the object.
C. a real image.
D. an virtual image.
E. None of these statements are always true.
Physics 231 Fall 2018
October 5th, 2018
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Midterm 1
Name:
9. (5 points) Light of wavelength λ passes through a single slit of width a and forms a
diffraction pattern on a viewing screen. If this light is then replaced by light of wavelength
2λ, the original diffraction pattern is exactly reproduced if the width of the slit
A. is changed to 41 a.
B. is changed to 12 a.
C. is changed to 2a.
D. is changed to 4a.
E. remains the same - no change is necessary.
10. (5 points) A miniature spaceship is flying past you, moving horizontally at a substantial
fraction of the speed of light. At a certain instant, you observe that the nose and tail
of the spaceship align exactly with the two ends of a meter stick that you hold in your
hands. Rank the following distances in order from longest to shortest:
(i) the proper length of the meter stick;
(ii) the proper length of the spaceship;
(iii) the length of the spaceship measured in your frame of reference
(iv) the length of the meter stick measured in the spaceship’s frame of reference.
A. (i) = (ii) > (iii) > (iv)
B. (ii) = (iv) > (i) = (iii)
C. (i) = (iii) > (ii) > (iv)
D. (ii) > (i) = (iii) > (iv)
E. (i) > (ii) = (iv) > (iii)
11. (5 points) The captain of spaceship A observes enemy spaceship E escaping with a
relative velocity of 0.48c. A missle M is fired from ship A, with a velocity of 0.72c
relative to ship A. The relative velocity of approach of missle M, observed by the crew
on ship E, is closest to:
A. 0.37c
B. 0.27c
C. 0.34c
D. 0.30c
E. 0.24c
Physics 231 Fall 2018
October 5th, 2018
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Midterm 1
Name:
12. (5 points) Particle A, of mass mA = 1 (in suitable units) moving at speed vA = 0.8c
relative to the lab, collides head-on with particle B, of unknown mass mB , moving at
speed vB = 0.6c relative to the lab. After the collision, the particles are each moving
perpendicular to the original direction of motion. For this to happen, what must be the
rest mass of particle B?
A. mB =
B. mB =
C. mB =
D. mB =
E. mB =
4
3
16
9
3
4
16
15
20
9
13. (5 points (bonus)) An inertial frame S 0 is moving with a velocity of 0.8c relative
to the
√
inertial frame S in the x-direction. A particle in S has velocity ~v = (0.5, 3/2)c. Find
the velocity of the particle in S 0 .
A. v 0 = c at θ = 120◦
B. v 0 = c at θ = 60◦
C. v 0 = 0.92c at θ = 71◦
D. v 0 = 0.92c at θ = 109◦
E. v 0 = 0.5c at θ = 30◦
Physics 231 Fall 2018
October 5th, 2018
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