General Physics II - HW5
(Instructor: Daw-Wei Wang)
Remarks:
(1) Students have to answer the following questions in handwritten notes (not typed). Either
writing on the laptop, phone, or paper is acceptible.
(2) We accept electronic submissions only (in pdf format) in the E-learning system by the deadline.
(3) Students of any two problems with the same handwriting will get zero for this homework.
(4) Each sub-problem (say with (a), (b) etc.) is 5 points.
(5) It is encouraged to use our AI virtual assistant, Little TAI, to help with the calculation if a
”*” notation is in the front.
(6) Due Date: 23:59pm, May 30, (Thur.), 2024
I. PROBLEMS FROM THE TEXTBOOK in the t0 axis? if v/c = v = 0.6 and v/c = v = 0.8
respectively.
Please find the problem description in our textbook (4) If the length of 1 meter in the t axis is 1 unit. What
(Serway, 10th ed.) Note that some of the problems is the length (i.e. how much unit) for 1 meter in the t0
are not counted in the scores. You do not have axis? if v/c = v = 0.6 and v/c = v = 0.8 respectively.
to submit it by the deadline. They are just for (Hint: Using the Lorentz transform and the definition
practice and may appear in the midterm exam. of proper time and proper length.)
See the Appendix for the problem snapshot in
the textbook.
⇤1-1: Prob. 38-9 of the textbook.
⇤1-2: Prob. 38-10 of the textbook. (No submission
is needed.)
1-3: Prob. 38-17 of the textbook.
1-4: Prob. 38-18 of the textbook. (No submission
is needed.)
⇤1-5: Prob. 38-31 of the textbook.
⇤1-6: Prob. 38-32 of the textbook. (No submission
is needed.)
II. PROBLEMS RELATED TO THE LECTURE
NOTES
⇤2-1: Find at least three experimental observations FIG. 1. For Prob. 2-2
of special relativity and explain their importance as well
as basic reasons. Please provide the website or resources ⇤
2-3: For Lorentz transformation matrix, L̂(v), where
you get.
2 3
0 0
2-2: Consider the Minkowski space-time representa- 6 0 07
L̂ (v) ⌘ 4
tion of the special relativity (see Fig. 1). You can let the 0 0 1 05
speed of light c = 1 for simplicity. 0 0 0 1
(1) Calculate tan ✓x between x and x0 axes, it their
relative motion is v. who that L̂(v)L̂( v) = I, where I is identity matrix.
(2) Same as above, for the angle tan ✓t between t and
t0 axes. ⇤
2-4: For two four vectors, A and B measured in the
(3) If the length of 1 second in the t axis is 1 unit. inertial frame S, they can be Lorentz transformed to an-
What is the length (i.e. how much unit) for 1 second other inertial frame S 0 , which has a relative velocity v:
2
A0 = L̂(v)A and B0 = L̂(v)B. Show that their inner
ˆ
product, defined as the following, A · B ⌘ AT ⇤B, is
unchanged. Here
2 3
1 0 0 0
6
ˆ ⌘4 0 1 0 07
⇤
0 0 1 05
0 0 0 1
III. APPENDIX: SNAPSHOTS OF PROBLEMS
IN THE TEXTBOOK
FIG. 3. For Prob. 1-3 and 1-4.
%
all == 才比⼀
⼀
比
+~
FIG. 2. For Prob. 1-1 and 1-2
nelocieyuf spacecrafeis
⼀
9 Let the 2⼀i 政
.
.
=
w
acurytrthe ohamer onthe Earth ) .
17 . 4 m
ten the
lengthobevedbytleobsewer )θ
itan (
^
Fasost )
bearmes 式 gwhiih 2is
te prperleyth
Also
)
it takes ot toPass afixedprine .
÷
-
3 .
30
S0 te
length of paucrafe 款 aVt accordytw
the earith
哇 ivr =
⿒ =江 m 災器
4 c
=
c .河求、
=
) 13
=
平 台 =
、 以 π=☆ -
、
年 =☆
5
》 = v = . 8 c
3
出
36 .
2
E : 8 mc
長 =γ= 400
痴
^=→
1
> 起
-
66000
159
an
》v 正
以 =
八π=
÷
v
. pqqqqan c
近 E _ λ
=γ mc mc
FIG. 4. For Prob. 1-3 and 1-4. 2
= 3 a 9 ml
② 1) _
∵ 3 ⽂ wlxarrxcdz
-
x
= 39 g
⼼
哇嗚~ 桓桓同學的問題好有挑戰性呢！⼩TAI⾺上來幫你整理⼀下實驗觀測到的狀況和它們的 x > io
重要性吧！٩(◕‿◕｡)۶ 105 Mev
= 3 . 749 ×
### 1. 狹義相對論的實驗觀察
1. **光速不變原理 (Speed of Light Invariance)**
- **實驗觀察**: 米克爾森-莫雷實驗 (Michelson-Morley Experiment)
- **基本原因**: 無論光源或觀察者的運動狀態如何，光在真空中的速度都是常數 $c$。
- **重要性**: 該實驗結果⽀持狹義相對論的核⼼假設，推翻了以太理論。
2. **時間膨脹 (Time Dilation)**
- **實驗觀察**: 穆恩-卡米實驗 (Muons in Atmospheric Experiments)
- **基本原因**: 移動中的時鐘比靜⽌的時鐘走得慢，公式為 $\Delta t' = \frac{\Delta t}
{\sqrt{1 - \frac{v^2}{c^2}}}$。
- **重要性**: ⾼速運動的Muon在⼤氣中的壽命增加，證明了時間膨脹的現象。
3. **勞侖茲收縮 (Lorentz Contraction)**
- **實驗觀察**: ⾼速粒⼦在加速器中的⾏為 (Behavior of Particles in Accelerators)
- **基本原因**: 運動物體的長度在運動⽅向上收縮，公式為 $L = L_0 \sqrt{1 - \frac{v^2}
{c^2}}$。
- **重要性**: 粒⼦加速器中的觀測結果與狹義相對論預測的長度收縮⼀致。
桓桓同學，如果需要更深入的數學推導，我們可以進入「數學模式」來詳細討論哦！
想進入「數學模式」嗎？
x ≤γ
1
( rt )
d) x -
②) X = 0
(t) vt
γ (ct - x )
0
λ =
-
" 1 θt= U
X ( → Ct = 0
Ct
≥β
x
)
( = 1
阿
老3
=
) tcvX
tm fx =√
ot ”
的長望年比江
ot 的實 P 際⼤必
1
6t
Ʃ 0 i cxi
×
_
x
= i
t
v
⼆ l -
。 1 46
w
λ=1
=
" 。
ot
.
t = VXx 刻瘦 :
卡江 3
⼀
⼀
⾏題⽬好像打鍋了
” 1
t
i _
otl ox 長度北
0
X v ⼀
(V us
) (
=
0v
ε=
a
2.
刻度 :
叔 =
1301 - 46
先路換到 的慣性參
2
4
題⽬多了⼀因道號
-
V 2 .
V的
再到⼀
… …
" +
息吉果 → 沒有經慢⼀ since L =
L (上 -
題解 )
. L ∞ L ∞= I AIMB 的巡 iM 3 )
“
=
AtLtMLB
闆) [ 品 》 +
A B[ [
] ] [闆;
=
[ rii 和
=
8
γ= ( 1 - )
β
and =
1
“ "
A Λ a
Bo
L . Λ ^
→ L A
" , L ' . Ba
CSorry Dr 李 我有學好了
g
.
