Chapter 2: Kinematics

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14 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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Chapter 2: Kinematics


2.1 Uniform Motion


2.2 Instantaneous Velocity


2.3 Finding Position from Velocity


2.4 Motion with Constant Acceleration


2.5 Free Fall


2.6 Motion on an Inclined Plane


2.7* Instantaneous Acceleration

Stop to think 2.1 P38

Stop to think 2.2 P44

Stop to think 2.3 P48

Stop to think 2.4 P54

Stop to think 2.5 P61



Example 2.3 P 40

Example 2.4 P 41

Example 2.7 P 45

Example 2.10 P 47

Example 2.14 P 53

Example 2.16 P 56

Example 2.18 P 58

Motion in one dimension



Determining the signs of position, velocity and acceleration

1 s

2 s

3 s

4 s

x

Origin


(x=0)

10

cm

20

cm

40

cm

70

cm

Motion along a straight line

Can be illustrated by
position
-
versus
-
time graph:

x

t(s)

1

2

3

4

Continuous
(smooth) curve

Position vs time graphs

Interpreting a position graph

1.What is the position at t =0min

2.What is the position at t =30min

3.What is the velocity at t = 20min

4.What is the velocity at t = 50min

5. What is the acceleration at t=20min

6. If this is V vs. t graph, and x = 0 km

at t = 0min. What is the position at t = 80 min


Finding velocity from position graphically

Uniform Motion


V(avg)= comstant


The position
-
vs
-
graph
is a straight line


Vs =
∆s/ ∆t


S
f

= S
i

+ Vs ∆t



Instantaneous velocity


Using motion diagrams and graphs

dt
ds
t
s
V
t
s






0
lim
Stop to think 2.2

Which velocity
-
versus
-
time goes with the
position

versus
-
time graph

C

Relating a velocity graph to a position graph

T

The value of the velocity at

Any time equals the slope of

The position graph


Using calculus to find the velocity

Ex. A particle’s position is given by the function



1.What is particle’s position at t = 2s?


x =
-
8+6 =
-
2 m


2. What is the velocity at t = 2s





V|
t=2

=
-
3(2)
2
+3=
-
9 m/s

m
t
t
x
)
3
(
3



2
( 3 3)/
dx
V t m s
dt
   
Finding position from Velocity




tf
ti
s
i
f
dt
V
S
S
tf
and

between ti

V

curve
volocity
under the

area
s


i
f
S
S
Example 2.9

1.Where is particle’s turning point?

2.At what time does the particle reach the origin?

Motion with constant acceleration

t
V
V
t
V
a
i
f






t
a
V
V
i
f



2
)
(
2
/
1
2
)
(
2
)
(
t
a
t
V
t
V
t
a
V
t
V
V
s
i
i
i
i
f













s
a
Vi
V
a
V
V
V
V
t
V
V
s
f
i
f
i
f
i
f











2
)
(
2
)
(
2
)
(
2
2
Definition of acceleration

See page 57

If set t0=0s,
∆t = t

Example 2.13

A rocket sled accelerates at 50m/s
2

for 5.0 s. Coasts for 3.0 s, then deploys a
parachute and decelerates at 3.0m/s
2

until coming to a halt.

What is the maximum velocity of the rocket sled?

What is the total distance traveled?

The apple and feather in this photograph are falling in a vacuum

Two objects dropped from the same height will, if air resistance can be neglected

Hit the ground at the same time and with the same speed


Free Fall

downward
y
verticall
,
)
(
g
freefall
a


g = 9.8m/s
2


If we choose the y
-
axis to point vertically up

g
freefall
a


)
(

Example 2.16 A falling rock

A rock is released from rest at the top of 100
-
m
-
tall building.

How long does the rock take to fall to the ground, what is impact
velocity?


Y
0
=100m Y
1
=0m


V
y0
= 0 m/s t
0

= 0 s


2
0
1
2
/
1
gt
y
y


s
g
g
y
y
t
52
.
4
)
0
100
(
2
)
(
2
1
0





s
m
gt
V
V
y
y
/
3
.
44
52
.
4
8
.
9
0
1







Motion on an inclined plane


sin
|
|
g
a
s

Instantaneous Acceleration

dt
dV
t
V
a
t






0
lim



tf
ti
i
f
adt
V
V
Homework 2.50


A 1000Kg weather rocket is launched straight up. The rocket motor
provides a constant acceleration for 16s, then the motor stops. The
rocket altitude 20 s after launch is 5100m. You can ignore the air
resistance

a) What was the rocket’s acceleration during the first 16 s.

The rocket launched with Vo = 0, after 16 s



b) After motor stops, the acceleration is

g as free fall.





c) The rocket’s speed as it passes through a cloud 5100m above the ground

a
at
V
16
0
1



a
at
y
128
2
/
1
0
2
1



2
2
1
2
1
1
2
4
8
.
9
2
/
1
4
16
128
)
)
(
2
/
1
(










a
a
t
t
g
t
v
y
y
2
/
27
5100
4
.
78
192
s
m
a
a




s
m
a
t
t
g
V
V
/
392
4
8
.
9
16
)
(
1
2
1
2







Quiz questions:

Two stones are release from rest at certain height one after
the other


A) Will the difference in their speed
increase, decrease or stay the same


B) Will their separation distance increase,
decrease or stay the same