Kinematics What is Kinematics? Focus Velocity and Acceleration

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13 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

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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

body frame
y
y
x

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

扨 汨 b d (
•
L
整e

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