等加速度運動 - ocw.nctu.edu.tw

交通大學李威儀

p. 1

等加速度運動

Uniformly

Accelerated

Motion


p. 2

十七世紀後 ,

物理研究的快速發展與數學密不可分

從伽利略和牛頓開始 ,

物理與數學就緊密的結合在一起了

這一章就將示範

如何將物體運動中我們所關心的幾個參數之間的關係

以數學的形式串聯在一起

Kinematics, 運動學描述運動現象中幾個參數彼此之間的關係

Dynamics, 運動力學研究運動狀況為何改變


p. 3

運動方式 1. Translational 移動

2. Rotational 轉動

3. Vibrational 振動

for translational motion :

4 vectors position 位置

( 四個要考 displacement 位移

慮的向量 ) velocity 速度

acceleration 加速度

簡介 ( Introduction )


p. 4

• average speed, a scalar

speed = (distance traveled) / (time elapsed) = DS / Dt

• position vector : r = x i + y j + z k •

• displacement vector : Dr = rf – ro

• average velocity : vav = ( rf – ro ) / ( tf – to ) = Dr / Dt

( to : 起始時間 , tf : 終止時間 )

x

y

ro

rf

Dr

Ds

速率 ( Speed ) 與速度 ( Velocity )

等速率圓周運動 :

速率 ( speed ) 是定值

速度 ( velocity ) 則隨時在變


p. 5

• average velocity : vav = ( rf – ro ) / ( tf – to ) = Dr / Dt

• In one-dimensional ( 一維空間 ) problems :

vav vav = Dx / Dt ( m/s ) = ( xf – xo ) / Dt

上例中 , different path, but same vav

t (s)

x (m)

to tf

xf

xo

Dx|

Dt

向量可以簡化為純量問題來考慮 :

Dx > 0 vav > 0

move in positive x direction

Dx < 0 vav < 0

move in negative x direction


p. 6

• Instantaneous velocity

1-dim : vx = lim Dx / Dt = dx / dt

( lim Dx dx, lim Dt dt )

vx : slope of the tangential line

vx > 0 +x 方向運動

vx < 0 -x 方向運動

2-dim and 3-dim : v = lim ( Dr / Dt ) = d r / dt

v : in the direction tangential to the path

Dt0

Dt0 Dt0

t

x

to tf’ tf

vav

vx

(Dt0)

Dt0


p. 7

v = d r / dt

= d ( x i + y j + z k ) / dt

= (dx / dt) i + (dy / dt) j + (dz / dt) k

= vx i + vy j + vz k

vx = dx / dt, vy = dy / dt, vz = dz / dt

v

circular motion

圓周運動


p. 8

position change velocity (d r / dt = v )

velocity change acceleration

average acceleration :

aav = Dv / Dt (m/s2)

= ( vf – vo ) / ( tf – to )

instantaneous acceleration :

a = lim ( Dv / Dt ) = dv / dt

= ( ) =

Dt0

d dr d2 r

dt dt dt2

加速度 ( Acceleration )


p. 9

ax 與 vx 同號速率加快

ax 與 vx 異號速率減慢

vx

t

t

ax

vx

t

t

ax

ex. ( one- dim. )


p. 10

• in previous sections :

from x vs. t v

v vs. t a

in this section :

from a vs. t v

v vs. t x

• a = dv / dt dv = a dt

( Dv = a Dt )

等加速度運動的例子 :

( 真空中的 ) 自由落體

線性等加速度運動 ( Linear Motion Under Const. Accel. )

• 描述一維空間的線性運動時

可以直接以正負值來代表向量的方向


p. 11

• a = dv / dt dv = a dt ( Dv = a Dt )

v(tn) – v(t=0)

= Dv1 + Dv2 + …… + Dvn

= a(t1) Dt + a(t2) Dt + ….. + a(tn) Dt

= a(tm) Dt = Dvm

= area under the ( a vs. t) curve

= a dt = dv

dv = a dt

v(tn) – vo = a tn

v(t) = vo + at

t

a

t

v

vo

S S n

m=1

n

m=1

tn

0

v

vo

v

vo

tn

0

t1 t3 tn-1 t2 t4 tn

vo

a t


• v = dx / dt dx = v dt ( Dx = v Dt )

similarly x – xo (@ t = 0)

= S Dx = S v Dt

dx = v dt

dx = (vo + at) dt

(area under the v vs. t curve)

= vo dt + a t dt

x – xo = vo t + ½ a t 2

x

xo

t

0

x

xo

t

0

t

0

t

0

t

v

vo

a t


p. 13

• 面積分也可以為負值

e.g.

• a = dv/dt = (dv/dx) (dx/dt) = v (dv/dx)

vdv = adx

v dv = adx = a dx

(vn /2) – (vo /2) = a (xn – xo)

vn – vo = 2a (xn – xo)

v2 – vo = 2a (x – xo)

a

t

adt = a t < 0 t

0

vn

vo

xn

xo

xn

xo

2 2

2 2

2


p. 14

三秒鐘 , 他可以加速

到100 公里的時速

+ y

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html


p. 15

ex. James Bond is walking under the tower of Pisa (54.6m high).

Somebody throws down a 10m long and wide board. Some girl

screams as a warning after 1.34 sec. Can Bond escape safely?

y = yo + voyt + ½ gt2

54.6 = ½ (9.8) t2 t = 3.34 sec.

Bond has 2 sec. to escape

x = xo + voxt + ½ at2

10 = ½ at2 = ½ a 22 = 2 a

a = 5 m/s2 at least

With Bond’s fast response,

maybe he can escape, as usual.


p. 16

• Terminal speed of free-falling body in air

for a person

vT ~ 300 km/hr ( 垂直降下 )

vT ~ 200 km/hr ( 四肢張開 )

vT

t

v in vacuum

in air


p. 17

• 自由落體的真正「終極速度」


p. 18

• Galileo proposed :

a 2-dim. motion 2 independent motions

水平 - 無外力作用，等速度運動

垂直 - 等加速度運動

如何以數學的形式描述這樣的運動 :

1st, choose a ref. frame : assume xo = 0

x = voxt , vy = voy – g t

y = yo + voy t – ½ g t2

vy = voy – 2g ( y – yo )

( read the text for examples )

2 2 x

y

vox ( voy = 0 )

拋體運動 ( Projectile Motion )


p. 19

Projectile Motion


p. 20

Projectile Motion


p. 21

This astonishing flying ring has amazingly

straight and accurate long flight.

This disk holds the Guinness world record for the

world's farthest thrown object –

an incredible 1,257 feet.

Inventor, Alan Adler

Stanford Professor


p. 22

例如 :

非等加速度運動 ( Non-Constant Acceleration Motion )

v(tn) – vo = a dt tn

0

x(tn) – xo = v dt tn

0

vo : t = 0 時的起始速度

xo : t = 0 時的起始位置


p. 23

非等加速度運動 ( Non-Constant Acceleration Motion )

等加速度運動 - ocw.nctu.edu.tw

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