課程名稱︰普通物理學甲上
課程性質︰系必修
課程教師︰張寶棣
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰104/1/15
考試時限（分鐘）：160
試題 :
1. Please answer the following questions.
‧List the differences of Doppler effect between mechanical waves and
electromagnetic waves. How do we define the past, present and future
of a given event in space-time diagram? (5%)
‧How do you reconcile the principle of conservation of energy with
the observation that, in one region, the wave medium is not in motion
when waves interfere destructively? (5%)
‧The relative velocity between the A frame and B frame is in the
x-direction and their origins coincide at some time. Please explain
why the y and z positions of an event are the same observed in both
frames but v_y and v_z are not.
2. A Feynman clock consists of two perfectly reflecting plan parallel mirrors
which are separated by a fixed distance, which face each other, and between
which a very short pulse of light is reflected back and forth. The time
between successive reflection of the pulse at one of the mirrors is the
unit of time for an observer at rest relative to the mirrors. Figure 1 is
a diagram of a Feynman clock as seen in the A-frame, relative to which the
clock is at rest. The diagram shows the world line of a fixed point on each
of the mirrors and the world line of a light signal which is reflected from
the lower mirror at event E0, which strike the upper mirror at event E1,
and which is reflected again from the lower mirror E2. The light signal
travels along the x-axis. The time interval between two successive
reflections from the lower mirror is ΔtA, and the distance between the
mirrors is ΔxA.
(a) Construct a Loedel diagram from fig. 1 in which the B-frame moves
backward along the xA-axis a speed v of about 0.5c.
(b) Show that the time interval between E0 and E2 in frame B is given by
the time dilation formula ΔtB = ΔtA * 1 / √(1-(v/c)^2)
(c) Show that the B-time interval Δt1B between E0 and E1 and the interval
Δt2B between E1 and E2 are related to the corresponding A-time interval
by Δt1BΔt2B = Δt1AΔt2A. (15%)
　　　　 xA
　　　　　↑
　　　　　｜
　　　　　｜　　　　 E1 world line of a point
　　　－－＋－－－－－－－－－－－－－－－－－ on upper mirror
↑　｜　　　　／＼　　　　　　　　／
　　　｜　｜　　　／　　＼　　　　　　／
ΔxA｜　　／　　　　＼　　　　／world line of light
　　　｜　｜　／　　　　　　＼　　／
　　　↓　｜／45°　　　　　　＼／
　　　－－＋－－－－－－－－－－－－－－－－－ world line of a point
　　　　E0｜←－－－cΔtA－－－→E2 on lower mirror
　　　　　｜
　　　　　｜
　　　　　－－－－－－－－－－－－－－→ctA
Figure 1
3. A couple of astronauts took a spacecraft to a certain galaxy. When they
left the Earth with a constant speed of 0.95c, the female astronaut found
that she was pregnant. After 10 months spacecraft time, a male baby was
born and the male astronaut immediately sent out the message to the Earth
with speed of light to apply for a birth certificate. If we neglect the
acceleration time that the spacecraft took off, how far was the spacecraft
observed by the Earth people when the baby was born? When did the
application signal reach the Earth according to the city government? If
the date that the couple left is March 2nd, 2040, what is the birthday of
the baby in the Earth time? (15%)
4. A standing wave pattern on a string is described by
y(x,t) = 0.04sin(5πx)cos(40πt), (1)
where x and y are in meters and t is in seconds.
(a) Determine the locations of all nodes for 0.00 ≦ x ≦ 0.40m. (3%)
(b) What is the period of the oscillatory motion of any (nonode) point on
the string. (3%)
(c) What are the speed and the amplitude of the two traveling waves that
interfere to produce this wave? (6%)
(d) At what times for 0.000 ≦ t ≦ 0.050s will all the points on the
string have zero transverse velocity? (3%)
5. Hovering over the pit of hell, the devil observes that as a student falls
past (within terminal velocity), the frequency of his scream decreases
from 842 to 820 Hz.
(a) Find the speed of descent of the student.
(b) The student's scream reflects from the bottom of the pit. Find the
frequency of the echo as heard by the student.
(c) Find the frequency of the echo as heard by the devil.
Assume that the speed of sound is 343 m/s. (10%)
6. Sinusoidal waves are generated on an infinitely long rope. One wave,
y1(x,t), moves to the left and has amplitude y0, wave number k1, and
angular frequency ω1; the other, y2(x,t), moves to the right with
amplitude y0, wave number k2, and angular frequency ω2. Each wave has
the same phase.
(a) Express yi(x,t) for each of the waves.
(b) Assuming that the rope is a nondispersive medium, what is ω2 in
terms of k1, k2, and ω1?
In parts (c) and (d), assume that k2=k1+δk and ω2=ω1+δω. Take the
speed of traveling waves on the rope to be v.
(c) What is δω in terms of δk and v?
(d) Superpose these waves, assuming that δk and δω are small. Use
the notation kav = (k1+k2)/2 and ωav = (ω1+ω2)/2. Show that the result
of the superposition is two waves, one moving at the low speed
vl = vδk/2kav and wavelength 2π/kav, and one moving at the high speed
vr = 2vkav/δk and the long wavelength 4π/δk. (20%)
7. Show how it is possible for one who knows the A- and B-coordiate intervals
between two events to construct the corresponding Loedel diagram and from
it find geometrically the relative speed between the A- and B-frames. (5%)
8. The star Arcturus is 36 light-yr from the earth (1 light-yr=9.463*10^15 m).
At what speed must a space ship travel relative to the earth so that this
distance is shortened to 1.0 light-yr as measured in the rest frame of
the ship? What time, as measured by a clock carried in the ship, is needed
for this trip from earth to start? (5%)