Rectilinear Motion

1

O

Reference Path Frame

(x,y) coord

r

q

(r,q) coord

vx

vy

x

y

v

x

 x  v

_y

 y 

vn

n

0

v  v

_t

 v

r

vr

v_q

v

_q

 r q  v

r

 r 

Path Reference

Frame

ax

ay

x

y

at

an

r

ar

a_q

vt

a

x

 x  a

^y

  y

2 n

a v



 a

_t

 v 

2

a_q  rq  rq

2

ar  r r

q



(n,t) coord

velocity meter

Summary: Three Coordinates (Tool)

Velocity Acceleration

Observer Observer’s

measuring tool

Observer

2

O

Reference Path Frame

(x,y) coord

r

q

(r,q) coord

vx

vy

x

y

v

x

 x  v

_y

 y 

vn

n

0

v  v

_t

 v

r

vr

v_q

v

_q

 r q  v

r

 r 

Path Reference

Frame

ax

ay

x

y

at

an

r

ar

a_q

vt

a

x

 x  a

^y

  y

2 n

a v



 a

_t

 v 

2

a_q  rq  rq

2

ar  r r

q



(n,t) coord

velocity meter

Choice of Coordinates

Velocity Acceleration

Observer Observer’s

measuring tool

Observer

4

Path

(x,y) coord

r

q

(r,q) coord (n,t) coord velocity meter

Translating

Observer

Two observers (moving and not moving) see the particle moving the same way?

Observer O (non-moving) Observer’s

Measuring tool

Observer (non-rotating)

Two observers (rotating and non rotating) see the particle moving the same way?

Observer B (moving)

Rotating

No!

No!

“Translating-only Frame”

will be studied today

Which observer sees the “true” velocity?

both! It’s matter of viewpoint.

“Rotating axis”

will be studied later.

Point: if O

understand B’s motion, he can

describe the velocity which B sees.

This particle path, depends on specific observer’s viewpoint

“relative” “absolute”

A

“translating”

“rotating”

5

2/8 Relative Motion (Translating axises)

 A = a particle to be studied

r 

A

A

Reference frame O

frame work O is considered as fixed (non-moving)

r 

B

 If motions of the reference axis is known, then “absolute motion” of the particle can also be found.

O

 Motions of A measured using framework O is called the “absolute motions”

 For most engineering problems, O attached to the earth surface may be assumed “fixed”;

i.e. non-moving.

 Sometimes it is convenient to describe motions of a particle “relative” to a moving “reference frame” (reference observer B)

B

Reference frame B  B = a “(moving) observer”

B

r 

A_/

 Motions of A measured by the observer at B is called the “relative motions of A with respect to B”

6

Relative position

 If the observer at B use the x-y **

coordinate system to describe the position vector of A we have

j y i

x

r

_AB

ˆ ˆ

/

 



where

= position vector of A relative to B (or with respect to B), and are the unit vectors along x and y axes

(x, y) is the coordinate of A measured in x-y frame

i ˆ

^{A B}

ˆ j r 

_/

** other coordinates systems can be used; e.g. n-t.

r 

B

B

r 

A

B

r 

A_/

A

X Y

x y

O

ˆ j

i ˆ

 Here we will consider only the case where the x-y axis is not rotating (translate only)

J ˆ

I ˆ

7

( ˆˆˆˆ ) ( )

A B

r  r  xi  yj  xi  yj

ˆˆˆˆ ( )

A B

r  r xi  yj  xi  yj

     

r 

B

B

r 

A

/

r 

A B

A

X Y

x y

O

ˆ j

i ˆ

 x-y frame is not rotating (translate only)

Relative Motion (Translating Only)

Direction of frame’s unit vectors do not change

ˆ 0 i   

ˆ j  0 



0

/

v 

A B

/

aA B

0

/

A B A B

r    r  r 

xi ˆˆ  yj

Notation using when B is a translating frame.

B A B

A

v v

v     

_/

B A B

A a a

a     _/

Note: Any 3 coords can be applied to Both 2 frames.

8

Understanding the equation

B A B

A v v

v      /

Translation-only Frame!

Path

Observer O Observer B

This particle path, depends on specific observer’s viewpoint

r 

A

A

reference framework O

r 

B

O

B

reference frame work B

B

r 

A_/

A

/

v 

A O

/

v 

B O

Observer O Observer O

Observer B

(translation-only

Relative velocity with O)

This is an equation of adding vectors of different viewpoint (world) !!!

O & B has a “relative” translation-only motion

9

The passenger aircraft B is flying with a linear motion to theeast with velocity v_B = 800 km/h. A jet is traveling south with velocity v_A = 1200 km/h. What velocity does A appear to a passenger in B ?

A B A B

v   v   v 

Solution

B 800 v 

A 1200 v 

x y

vA B

q

jˆ 1200 iˆ

800

v_{A B}  

⁸⁰⁰ ^{2 1200}²

vA B  

1200 tanq  800

1200 ˆ

vA  j v_B  800iˆ

10

A B

A B

v  a 

A B A B

v  v  v

A B A B

a  a  a

18 ˆˆ 5 /

A 3.6

v  i  i m s

ˆ

2

3 /

a 

A

 i m s

2 3 1 rad/s 60 10

q      

q 0

Translational-only relative velocity

You can find v and a of B

11

v

_A

v

_B

v

_A/B

Velocity Diagram x

y

a

_A

a

_B

a

_A/B

Acceleration Diagram x y

 

9 ˆˆ

( ) cos 45 sin 45 2 2

10

o o

vB   i  j  i  j

/ 3ˆˆ 2 /

A B A B

v  v  v  i  j m s

 

2 cos 45^o ˆˆˆˆsin 45^o 0.628 0.628

B B

a v i j i j

 R     



/ 3.628ˆˆ 0.628 /

A B A B

a  a  a  i  j m s

v

rad/s 10

q  

9

B 10

v  rq  

q 0

2 B B

a v

 R

5ˆ /

vA  i m s _{ˆ 2}

3 /

aA  i m s

t 0

a  rq

2 2 n

a r v q r

  

: /

B A rel B A

v   v   v      r 

?

?

/

A B A B

v   v   v 

?

/

B A B A

v   v   v 

B

?

/

A B A B

v   v   v 

?

/

B A B A

v   v   v 

Yes

Yes

Yes

No O

Is observer B a translating-only observer

relative with O

13

 50^ v_{A B}

: obserber B, translating?

/ : obserber A, translating?

vB A

B

vA

To increase his speed, the water skier A cuts across the wake of the tow boat B, which has velocity of 60 km/h. At the instant when q = 30°, the actual path of the skier makes an angle  = 50° with the tow rope. For this position determine the velocity v

_A

of the skier and the value of

^q Relative Motion:

(Cicular Motion)

m 10

A B

A B

A B 10

v  rq  q

s m 67 . 6 16

. 3 v_B  60 

vA

60 120^ 20^ 40



 sin120

v 40

sin 67 .

16 _A



s m 5 . 22 v_A 

sin 20

16.67 10

sin 40

vA B  ^  q

 

0.887rad s q 

30

D

M

?

^O.K.

?

Point: Most 2 unknowns can be solved with

1 vector (2D) equation.

A B A B

v   v   v 

20^

20^ 60^

60^

30^

q 30^

Consider at point A and B as r-q coordinate system

14

2/206 A skydriver B has reached a terminal speed . The airplane has the constant speed

and is just beginning to follow the circular path shown of curvature radius = 2000 m

Determine

(a) the vel. and acc. of the airplane relative to skydriver.

(b) the time rate of change of the speed of the

airplane and the radius of curvature of its path, both observed by the nonrotating skydriver.

B 0 a  

50ˆ vA  i

A 0

a  

/ , /

B A B A

r q^



50 / vB  m s

50 ˆ v 

B

  j

 

aA x  ^{0 ( )}aA t

 

_{A y} ^{( )}_{A n} ^A2

A

a a v

  

ˆˆ

2

( ) 1.250 /

A y

a  a j  j m s

/

= - ,

/

-

A B A B A B A B

v  v  v  a   a  a 

50 m/s vA 

r

v

^r

50 m/s vB 

/ 50ˆˆ 50

vA B  i  j

/ 1.250ˆ aA B  j

15 /

v 

A B

/ aA B

(b) the time rate of change of the speed of the airplane and the radius of curvature of its path, both observed by the nonrotating skydriver.

r

v

^r

n

t

/ /

( ) sin 45^o

r A B t A B

v  a  a

2 /

/ /

( ) cos 45^o

A B

A B n A B r

v a a

 ^{ }

/

,

/

A B A B

v  a 

/ 50ˆˆ 50

vA B  i  j

/

1.250 ˆ a 

A B

 j

B 0 a  

50ˆ vA  i

50 ˆ v 

B

  j

ˆ

2

1.250 /

aA  j m s

coord n t 

v 

r



^r

45^o 45^o

16

1000 ˆ m /

A 3.6

v  i s

ˆ 2

1.2 /

aA  i m s

1500 ˆ /

B 3.6

v  i m s

0 / 2

aB   m s

, : relative world r q^



/ , /

B A B A

r q^



coord r q

/

,

/

B A B A

v  a 

17

/ 500ˆ

BA 3.6

v  i

/

1 .2 ˆ

BA

a    i a v

q

( v

_{B A r}/

)  r 

^{r v} 

cos q



120.3

(

v_{B A}/

)

_q  r

q



q

  

0.00579

2

(a_{B A r}/ )  r rq (a_{B A}/ )_q  rq 2rq

0.637

r  



0.166 10 3

q  ^ q

r

1000 ˆ m/

A 3.6

v  i s

ˆ 2

1.2 /

aA  i m s

1500 ˆ /

B 3.6

v  i m s

0 / 2

aB   m s

cos

v q



sin

v

q

 

cos

a q

 

sin

a q



30^o



coord r q

1800 1200 sin 30^o 1200

r   