1

KinematicsKinematics

Course Virtual Worlds

INFOVW - 2010

What is Kinematics?What is Kinematics?

• Kinematics is the study of the motion of bodies

without regard to the forces acting on the body

• Focus:

P iti

–

P

os

iti

on

–Velocity

–Acceleration

• How are these related and how do they change

over time?

FocusFocus

• We will focus on

–Particles

–Rigid bodies

•

A

syste

m

o

f

pa

r

t

i

c

l

es

t

h

at

r

e

m

a

in

at

fix

ed

d

i

sta

n

ces

syste o pa t c es t at e a at ed d sta ces

from each other with no relative translation or

rotation among them

Velocity and AccelerationVelocity and Acceleration

2

• Velocity is a vector quantity,

–Direction

–Magnitude

d f l ll d d

SpeedSpeed

• Magnitu

d

e o

f

ve

l

ocity is ca

ll

e

d

spee

d

t

s

v

Speed exampleSpeed example

1000 m

kph

t

s

v 72

s

m

20

s)1060(

m1000

1

t

1

= 10 s

2

t

2

= 60 s

Instantaneous velocityInstantaneous velocity

ds

s

dt

ds

t

s

v

t

)(lim

0

Integrate this…

Relationship between s and v

Relationship between s and v

dtvds

dt

ds

v

2

1

2

1

2

1

12

t

t

t

t

s

s

dtvsss

dtvds

3

Displacement and DistanceDisplacement and Distance

• In 1D (straight-line movement) displacement

and distance traveled is the same

•

I hi h di i thi i diff t

•

I

n

hi

g

h

er

di

mens

i

ons

thi

s

i

s

diff

eren

t

Show this on whiteboard…

AccelerationAcceleration

t

v

a

dt

dv

a

t

v

a

t

0

lim

Integrate this…

Speed change…

Speed change…

dtadv

dt

dv

a

2

1

2

1

2

1

12

t

t

v

v

t

t

dtavvv

dtadv

Constant Acceleration

Constant Acceleration

4

Constant AccelerationConstant Acceleration

• If an object experiences constant acceleration its

speed changes accordingly

–Example is the acceleration due to the earth’s

gravity (a = g = 9 81 m/s2)

gravity

(a

=

g

=

9

.

81

m/s2)

• This can be calculated by solving the following:

2

1

2

1

v

v

t

t

dtadv

Do this on whiteboard…

Velocity dependent on distance

Velocity dependent on distance

d

ds

d

dt

dv

a

dvvdsa

dv

dt

d

sa

Kinematic Diffential Equation of Motion

Integrate this…

ExamExam

• You should be able to calculate all these things

by heart

• Table 2.1 in the book will not be supplied at

the exam

NonNon--Constant AccelerationConstant Acceleration

5

NonNon--constant accelerationconstant acceleration

• This is very common

• For example, any object moving in a real world

will experience drag

d i l t l t

–more on

d

rag

i

n

l

a

t

er

l

ec

t

ure

• One type of drag is dependent on speed

2

vka

Resulting equation…

Resulting equation…

k

dv

vka

2

2

dtkdv

v

v

k

dt

2

2

1

Integrate this…

…integrated……integrated…

1

2

dtkdv

v

)1(

...

1

1

2

tkv

v

v

…substitute……substitute…

dsdtv

dt

ds

v

)1(

where

1

1

tkv

v

v

Integrate this…

6

…results in…results in

t

k

v

)

1ln

(

1

k

s

)

(

1

Question for you

Question for you

• When and where will an object stop under the

drag in the previous example?

•

U th f ll i l

•

U

se

th

e

f

o

ll

ow

i

ng va

l

ues

–k = 10

–v1 = 20 m/s

General casesGeneral cases

• In general very hard to calculate using these

formulas

• Usually solved by numerical integration

•

Will b di d i l t l t

•

Will

b

e

di

scusse

d

i

n

l

a

t

er a

l

ec

t

ure

2D Particle Kinematics2D Particle Kinematics

7

Independence of two directionsIndependence of two directions

• In the 2D case, you can regard the two

directions as being independent

–Two sets of 1D problems

y

x

y

x

a

a

v

v

y

x

a

v

s

Written out this leaves…Written out this leaves…

y

x

v

v

d

dy

dt

dx

dt

y

x

d

dt

ds

v

y

x

a

a

dt

yd

dt

xd

dt

d

dt

d

d

t

2

2

2

2

2

2

sv

a

An ExampleAn Example

v

1

=800 m/s

??

30°

assume: g = -10 m/s

2

3D Particle Kinematics3D Particle Kinematics

8

Just an extension of 2DJust an extension of 2D

• Nothing more complicated than 2D case

• Just add an extra dimension

•

R lt i

•

R

esu

lt

s

i

n:

–position: x, y, z

–velocity: v

x

, v

y

, v

z

–acceleration: a

x

, a

y

, a

z

Rigid Body KinematicsRigid Body Kinematics

Similar Kinematics

Similar Kinematics

• Rigid Body Kinematics is basically particle

kinematics with rotation

•

M t i t i t th t f

•

M

os

t

conven

i

en

t

i

s

t

o use

th

e cen

t

er o

f

mass as

the particle for linear kinematics

–track C.o.M. translation

–track rotation around C.o.M

Local Coordinate FrameLocal Coordinate Frame

Ω

body frame

y

y

x

Ω

world frame

x

9

Angular Velocity and AccelerationAngular Velocity and Acceleration

dt

d

dd

dt

d

dt

d

2

2

Integrate this…

Points on the ObjectPoints on the Object

• Points on the object move

• Combination of two motions:

–linear motion of CoM

angular motion around CoM

–

angular

motion

around

CoM

• Want to calculate this because you want to

know stuff about the points

–For example, how hard will two object hit each

other

Arc LengthArc Length

• Call c

p

the arc length for a point on the object

• Let r

p

be the distance between this point and

the axis of rotation

L

Ω

扨 汨 b d (

L

整e

Ω

b

攠e

h

攠慮e

l

攠e

h

攠e

b

橥捴潴慴j

d

(

楮i

牡摩慮猩

pp

rc

Angular and Linear Velocity

Angular and Linear Velocity

p

dt

d

r

dt

dc

pp

rv

Differentiate this…

10

Angular and Linear AccelerationAngular and Linear Acceleration

dt

d

r

dt

dv

p

p

ra

t

Tangential linear acceleration

Centripetal Acceleration

Centripetal Acceleration

• Besides the tangential linear acceleration, there

is also the centripetal acceleration of a point on

the object

•

Thi i di t d t d th i f t ti

•

Thi

s

i

s

di

rec

t

e

d

t

owar

d

th

e ax

i

s o

f

ro

t

a

ti

on

• This is what you ‘feel’ when you go through a

corner in a car or bus

Centripetal AccelerationCentripetal Acceleration

2

r

v

a

n

2

ra

r

n

2D vs. 3D

2D vs. 3D

• In 2D there is no problem in using these scalar

quantities for angular speed and acceleration

•

H i 3D thi i f bl d

•

H

owever,

i

n

3D

thi

s

i

s more o

f

a pro

bl

em, an

d

vectors need to be used

11

Linear Tangential VelocityLinear Tangential Velocity

r

ωv

rαa

rωωa

t

n

)(

Resulting Quantities for Point

Resulting Quantities for Point

• Remember, the object moves linearly as the

CoM moves

• Rotation add to the movement for points on

th bj t

th

e o

bj

ec

t

• Total motion of a point on the object is the

sum of the two motions

Show on whiteboard…

ExerciseExercise

• Car drives around a bend with 20 m/s

• The diameter of the turn is 20 meter.

• Questions:

1

Wh t i th l l ti?

1

.

Wh

a

t

i

s

th

e angu

l

ar acce

l

era

ti

on

?

2.What is the centripetal acceleration?

3.Assume that the gravitational acceleration is 10

m/s

2

, what is the amount of Gs experienced

by the driver and passengers?

Questions??Questions??

12

Next Lecture…Next Lecture…

• Topic of the next lecture:

–Forces

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