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Ch2 Kinematics in One Dimension

講者：許永昌 老師

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ContentsReview of Ch1Preview of Ch2ActionExercises (Gotten from R.D. Knight,

Instructor Guide)Homework

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Review of Ch1Motion diagrams.The concepts of position, velocity and

acceleration.Pictorial representation.Units.

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Preview of Ch2 (P34~P63)This chapter is focused on the mathematical

representation of one dimensional system.

drv

dt

0v v

f

ir vdt

dva

dt

f

iv adt

0a a

Kinematics: 只講運動型式 ( 果 ) ，不講原因 ( 原因要等到 Newton‘s Law 才講 ) 。

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Action (constant velocity, different directions and starting points)Purpose:

From a motion diagram to a position graph and a velocity graph.

Coordinate system.Objects and Actors:

1 student.A coordinate system.

Actions:Plot the motion diagrams, x-vs-t graphs and v-vs-t graphs of

the following situations: Origin left to right right end Right end right to left origin -x right end 若已經有 x-vs-t graph ，有時也可以反過來畫出 motion diagram 來得到一點

感覺也不錯。The mathematical representation of a constant velocity.

O

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ExerciseDo A and B ever have the same speed? If so,

at what time or times?(1)

A

B(2)

A

B

t

x

t

x

1 2 3 4 5 1 2 3 4 5

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ExercisesDoes vx=x/t?Does the slope have unit?What are wrong in these equations?

dx=x2

vx=xv=-1 m/s

以上是常見的錯誤。

t1 2 3 4 5

x

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Exercise( 請預讀 p34~p38)

Rank in order the speeds at points A, B and C, from the fastest to the slowest.A verbal description of this motion.Rank themDraw a velocity-versus-time graph with a

proper numerical scale.

0 1 2 3 40

1

2

3

4

5

6

A

BC

t (second)

x (m

)

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Instantaneous velocity and derivative ( 請預讀 p38~p44)

Instantaneous velocity:

Mathematics:

0lim , for previous pages.st

s dsv s x

t dt

1 ,

.

.

nndt

ntdtd dA dBA B

dt dt dtd cA dA

cdt dt

Q: Do you know how to prove them?

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Summary IWe can get the velocity form the SLOPE of x-

vs-t graph.You need to make sure your definition about

the slope is correct.Before you are familiar in x-vs-t graph, I hope

you plot a motion diagram to help yourself to analyze the motion described by this x-vs-t graph.

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HomeworkStudent Workbook: 2.1.b2.42.52.62.10

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Action II( 請預讀 p44~p48)

Purpose: Realize the area under the v-vs-t curve is

the change of the position (displacement). i.e. Ds=the area under the v-vs-t curve.

Action:Example: A train is moving at a steady 30 m/s.

At t=0, the train’s engine is at x=0 m. Find the engine’s positions at t=1 s and t=2 s. Draw the x-vs-t graph and v-vs-t graph. Check the area under the v-vs-t curve.

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ExercisesPlease write down the expanded form of

What is wrong in this equation?

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1

.s kkk

v t

.f

i

t

x tx v dt

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The integrals

Mathematics:

0

1

The integral of from to lim ,

where .

f

i

Nt

s i f s s kt tk

f i

v tt t v dt v t

N t t t

1

,1

,

. Besides,

f

f

i

i

f f f

i i i

f f f i

i i i f

tnt n

tt

t t t

t t t

t t t t

t t t t

tt dt

n

A B dt Adt Bdt

cAdt c Adt Adt Adt

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Finding position from velocity 1 2 ...

f

i

t

t

dss s s ds

dt dt

f f

i i

t t

st tv dt

Ds1 Ds2 Ds3 Ds4 Ds5

ti tf

Ds

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ExerciseFind the position at times 1, 2, 3, 4 and 5

seconds. Assume x0= 0 m.

Draw a position-versus-time graph.(1)Change x0. (2)Change vx0 and the slope.

0 1 2 3 4 50

5

10

15

20

25

30

35

40

t (second)

v x (

m/s

)

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Summary II透過對速度 (velocity) 積分可以得到位移

(displacement) 。此積分對應到 v-vs-t 圖中線段下所圍面積。

STOP TO THINK: 這面積的單位？此積分符號中，請勿忽略 dt ，因為

。dt Dt 。面積 高寬

t1 2 3 4 5

vx

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Example 2.2 (p37)Purpose:

Verbal representation pictorial representation & graphs illustrate the problem-solving strategy

Bob leaves home in Chicago at 9:00 A.M. and travels east at a steady 60 mph. Susan, 400 miles to the east in Pittsburgh, leaves at the same time and travels west at a steady 40 mph. Where will they meet for lunch? Model:

Visualize:

Solve:

Assess:Please fill out a Dynamics Worksheets with me.

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HomeworkStudent Workbook:2.12

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Action III ( 請預讀 p48~p60)

Purpose:To build up the concept of the acceleration.直覺不一定對：位置一樣不代表 _______ ，速度為零不代表

______ ，有加速度不代表 __________ 。Objects:

1 toy car.1 incline.

Actions:Draw the motion diagram, y-vs-t and v-vs-t graphs of

a tossed toy car.Discuss the velocity and acceleration at the turning

point.

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constant acceleration For one dimensional case, the acceleration is

the slope of the velocity-versus-time curve.

STOP TO THINK: If a jet accelerates at 3.00 m/s2 during take-off, what are the velocities at t=1, 2, 3 second? (Without using any equations) y=0 m at t=0 s.

constant.v

at

t (s)

v y (

m/s

)

DtDvy

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Kinematics with constant acceleration (p52)CONDITION:

We can get (Please derive them by yourselves)

由 position, velocity & acceleration 的定義出發。 也請利用微分與積分在式子間做變換。 也請想想如何將它們與 graphs 上的斜率與面積對應。

constant.a

, ,

2,

2 2, ,

,

1,

2

2 .

x f x i x

f i x i x

x f x i x

v v a t

x x v t a t

v v a x

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Instantaneous acceleration ( 請預讀p61~p62)

Instantaneous acceleration:

0lim .t

v dva

t dt

.f

i

t

f i tv v adt

t (s)

v (m

/s)

a (

m/s

2)

t (s)

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ExerciseA sprinter accelerates at 2.5 m/s2 until

reaching his top speed of 15 m/s. He then continues to run at top speed. How long does it take him to run the 100-m dash?

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Summary III請多加練習在 motion diagram 與 x-vs-t, v-vs-t 與 a-

vs-t graphs 四者之間的轉換與關連，它們都是方便您分析問題的好工具。

有些觀念用問的我們知道錯，但是，這錯誤卻出現在我們的直覺裡；所以，善用上述 4 個工具，可避免這種錯誤發生。速度 =0 不代表加速度 =0 。有加速度不代表 speed 一定增加。

為此，這本書特別希望大家先1. 訂出座標2. 寫出 s, vs, as ( 即位置，速度與加速度在 s 方向的分量 ) 而不是

寫 r, v, a 。

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HomeworkStudent Workbook:

2.14, 2.15, 2.16, 2.18, 2.21, 2.22, 2.23Textbook’s Problems:

2.8, 2.9, 2.25, 2.27, 2.45, 2.49自己製作 p63 裡 terms and notation 裡頭的單字

卡片。