1

O

PathReference Frame

(x,y) coord

rq

(r,q) coord

xv

yv

x

y

xv xyv y

nv

0nv tv v

r

rvvq

v rq q rv r

PathReference Frame

xa

ya

x

y

ta

na

r

ra

aq

tv

xa x ya y2

nva

ta v

2a r rq q q 2ra r rq

(n,t) coordvelocity meter

Summary: Three Coordinates (Tool)Velocity Acceleration

Observer

Observer’smeasuringtool

Observer

2

O

PathReference Frame

(x,y) coord

rq

(r,q) coord

xv

yv

x

y

xv xyv y

nv

0nv tv v

r

rvvq

v rq q rv r

PathReference Frame

xa

ya

x

y

ta

na

r

ra

aq

tv

xa x ya y2

nva

ta v

2a r rq q q 2ra r rq

(n,t) coordvelocity meter

Choice of CoordinatesVelocity Acceleration

Observer

Observer’smeasuringtool

Observer

4

Path

(x,y) coord

rq

(r,q) coord

(n,t) coordvelocity meter

Translating Observer

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

Observer O(non-moving)

Observer’sMeasuring 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

Ar

A

Reference frame O

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

Br

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”

BAr /

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

6

Relative positionIf the observer at B use the x-y **

coordinate system to describe the position vector of A we have

jyixr BAˆˆ

/

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 framei j

BAr /

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

Br

B

ArBAr /

A

X

Y

x

y

O

j

i

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

J

I

7

ˆ ˆ ˆ ˆ( ) ( )A Br r xi yj xi yj

ˆ ˆ ˆ ˆ( )A Br r xi yj xi yj

Br

B

Ar/A Br

A

X

Y

x

y

O

j

i

x-y frame is not rotating (translate only)

Relative Motion (Translating Only)

Direction of frame’s unit vectors do not change

ˆ 0i

ˆ 0j

0

/A Bv

/A Ba

0

/A B A Br r r

ˆ ˆxi yjNotation using when

B is a translating frame.

BABA vvv /

BABA aaa /

Note: Any 3 coords can be applied toBoth 2 frames.

8

Understanding the equation

BABA vvv /

Translation-only Frame!Path

Observer O

Observer B

This particle path, depends on specific observer’s viewpoint

Ar

A

reference

framework O

Br

O

B

reference

frame work B

BAr /

A

/A Ov /B Ov

Observer O Observer O

Observer B (translation-onlyRelative 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 vB = 800 km/h. A jet is traveling south with velocity vA = 1200 km/h. What velocity does A appear to a passenger in B ?

A B A Bv v v Solution

800Bv

1200Av

x

y

A Bvq

j1200i800v BA

2 2800 1200A Bv

1200800tan q

ˆ1200Av j ˆ800Bv i

10

A B A Bv a

A B A Bv v v

A B A Ba a a

18 ˆ ˆ5 /3.6Av i i m s

2ˆ3 /Aa i m s

12 3 rad/s60 10

q

0q

Translational-only relative velocity

You can find v and a of B

11

vA

vB vA/B

Velocity Diagramx

y

aAaBaA/B

Acceleration Diagramxy

9 ˆ ˆ( ) cos45 sin 45 2 210

o oBv i j i j

/ˆ ˆ3 2 /A B A Bv v v i j m s

2

ˆ ˆ ˆ ˆcos45 sin 45 0.628 0.628o oBB

va i j i jR

/ˆ ˆ3.628 0.628 /A B A Ba a a i j m s

v rad/s

10q

910Bv rq

0q

2B

BvaR

ˆ5 /Av i m s 2ˆ3 /Aa i m s

0ta rq 2

2n

va rr

q

: /B A rel B Av v v r

?

?/A B A Bv v v

?/B A B Av v v

B

?/A B A Bv v v

?/B A B Av v v

Yes

Yes

Yes

No

O

Is observer B a translating-only observer

relative with O

13

50 : obserber B, translating?A Bv

/ : obserber A, translating?B Av

BAv

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 vA of the skier and the value of q

Relative Motion:(Cicular Motion)

m10A

B

A

B

10A Bv rq q

sm67.166.3

60vB

Av

60 120 20

40

120sinv

40sin67.16 A

sm5.22vA sin 2016.67 10sin 40A Bv q

0.887 rad sq

o30

D

M ? ?O.K.

Point: Most 2 unknowns canbe solved with 1 vector (2D) equation.

A B A Bv v v

20

2060

60

30

30q

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 mDetermine (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.

0Ba

ˆ50Av i

0Aa

/ /, B A B Ar q

50 /Bv m s

ˆ50Bv j

0 ( )A A txa a

2

( ) AA A ny

A

va a

2ˆ ˆ( ) 1.250 /A ya a j j m s

/ /= - , - A B A B A B A Bv v v a a a

50 m/sAv

rrv

50 m/sBv

/ˆ ˆ50 50A Bv i j

/

ˆ1.250A Ba j

15

/ A Bv/ A Ba

(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.

rrv

n

t

/ /( ) sin 45or A B t A Bv a a

2/

/ /( ) cos 45oA BA B n A B

r

v a a

/ / , A B A Bv a

/ˆ ˆ50 50A Bv i j

/ˆ1.250A Ba j

0Ba

ˆ50Av i

ˆ50Bv j

2ˆ1.250 /Aa j m s

coordn t

rv r45o

45o

16

1000 ˆ m /3.6Av i s

2ˆ1.2 /Aa i m s

1500 ˆ /3.6Bv i m s

20 /Ba m s

, : relative worldr q

/ /, B A B Ar q

coord r q

/ /, B A B Av a

17

/500 ˆ3.6B Av i

/

ˆ1.2B Aa i

va

q

/( )B A rv r cos 120.3r v q

/( )B Av rq q 0.00579q

2/( )B A ra r rq

/( ) 2B Aa r rq q q

0.637r

30.166 10q

rq

1000 ˆ m /3.6Av i s

2ˆ1.2 /Aa i m s

1500 ˆ /3.6Bv i m s

20 /Ba m s

cosv q

sinv q

cosa q

sina q

30o

coord r q

1800 1200 1200sin 30or