Kinematics: Intro to Motion

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13 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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Kinematics: Intro to Motion

Average Speed

Instantaneous Speed

Scalar vs. Vector measures

Graphing Vectors

Velocity vs. Speed

Acceleration

Objectives


Define average speed and its units of measure


Differentiate between average and instantaneous speed


Describe scalar vs. vector measures


Define displacement and describe why direction is an
important part of its definition.


Graph displacement using vectors


Add vectors to determine displacement


Velocity


distinguish between speed and velocity


Define acceleration and identify it’s units Relate velocity and
acceleration


Graph velocity and acceleration


Average and Instantaneous Speed


What are the two base units used to describe
speed?



Why is speed often described in terms of “average
speed”. Both average and instantaneous speed are
an “average” of a distance traveled over an amount
of time. How are they different?



If you get a speeding ticket and can prove your
average speed was below the “ticket speed” is that
enough evidence to win your case?

Definitions


Average Speed s = d/t


Average rate at which distance is covered over time


At any given instant the speed can be more or less than the
average


S = speed, d = distance t = time



Instantaneous speed


Time rate at which distance is covered at a given instant in
time.


Generally the average speed for a
very short time
interval
.




Problems


At 1:00 pm, a car, traveling at a
constant velocity of 94 km/h toward
the west, is 17 km to the west of your
school. Where will it be at 3:30 pm?




school

start

94 km/h for 2.5 h

94 x 2.5 = 235 km


17 + 235 km = 252 km

You travel 235 km…BUT your starting point is

17 km away from the school (on the same side that

you are traveling…so you start at 17, then ADD 235

km more

Problems


Suppose the car in Problem 7 started
at 17 km east of your school at the
same time, moving in the same
direction at the same velocity. Where
would it be at 3:30 pm. At what time
would it be at your school?

school

You travel 235 km…BUT your starting point is 17 km away from the school (and you
have to pass by the school (the first 17 km you travel), then continue on until you have
traveled a total of 235 km. Since 17 km is the distance you traveled BEFORE passing
the school 235
-
17 or 218 km is the distance you traveled to the west of the school.

Problems


At what time would you be at your
school?



school

Really this question is asking how long does it take you to travel 17 km if you are
traveling at 94 km/ hour



we travel 94 km/60 min so we will travel 17 km/ ? Minutes. Cross multiply to get
the answer: (60)(17) = 94(x). X = 10.85 or about 11 minutes.

Calculating the slope of a line

Position vs. Time

Time (s)

Position (m)

0

100 m

1

360 m

2

620 m

3

880 m

4

1140 m

5

1400 m

Calculating the slope

Distance Traveled
0
500
1000
1500
0
1
2
3
4
5
6
Time (sec)
Position (Distance), Km
Use the coordinates (x and y) of any two points.


Subtract the vertical distance between the two points (y2


y1)

Divide by the horizontal distance between the two points (x2
-
x1)

So…what is slope?


It is the



RISE


(height difference of the two points)


RUN (horizontal difference between two points)



The vertical axis is in what unit?

The horizontal axis is in what unit?


What is the name given to this “derived unit” and where did we see
it before?

Typical speed variation

Stoplight

Passing the truck

Behind a

slow truck

Turn onto

the highway

Time (minutes)

Speed

(km/h)

5 10 15 20

120



90




60




30

Scalar and vector measure


The mathematical quantities which are used to
describe the motion of objects can be divided into
two categories.


The quantity is either a
vector

or a
scalar
. These
two categories can be distinguished from one
another by their distinct definitions:



Scalars

are quantities which are fully described by a
magnitude (or numerical value) alone.



Vectors

are quantities which are fully described by both a
magnitude and a direction.


Vectors


Require that you know POSITION


Need a start (frame of reference)


Need a final position



Calculate DISPLACEMENT

Frame of Reference


A speedometer can tell us how far we
go (distance), but does not tell us
anything about where we started, in
what direction we traveled or our final
location.


In order to determine the position of an
object, we need to have a
reference
point

or a
frame of reference
.


Displacement


Where is the point of reference? Where is
the end point?


Trace the distance traveled


Identify the displacement of the walker from
the point of reference

Displacement and Distance


Distance is simply the “separation”
between two objects. It has only
magnitude

and is a
scalar quantity



Position has both
magnitude

and
direction

and is termed a
vector
quantity

Vectors:

Every Which Way

How do you calculate a distance vector?

Vectors:

Every Which Way

Where’s the Treasure?

Stevie Simple buys a
guaranteed treasure map
from Captain Jack
Sparrow. The map says to
walk 20 paces from Dead
Man’s Tree, turn and walk
5 more paces to the
treasure. Where is the
treasure?

Where’s the Treasure?

20 paces

5 paces

Treasure is

anywhere in

this ring

Tree

What’s Missing?


Directions!


Measurement with a direction is a
vector

Where’s the Treasure?

20 paces

5 paces

Tree

25 paces due East

5 paces due North

Treasure

Vector Representations


Represented as an

arrow



Magnitude

= scaled length of
arrow


Direction

is given by orientation
of the arrow


In an equation vectors are
represented as


Bold

variables




Variables with
a bar




Variables with
an arrow



y
y
y
http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/vectors/vd.html

What can you use vectors for?


Anything where
direction

matters


displacement



velocity


acceleration


force

Scalars


Physical quantities where direction
doesn’t

matter


mass



volume


energy


power

Vector Components

α

1
y
1
x
x

y


We can describe a vector by


its
magnitude

(or length) m


Its
orientation

given by angle
α

counterclockwise from the x
axis


OR


Its
components
or
projections

on the x and y
axes (x
1
, y
1
)


All vector diagrams need a scale

m

perpendicular

perpendicular

1 box=1 cm

Pythagorean Theorem

y


a, b and c form a
right
triangle


c is the
hypotenuse

2 2 2
c a b
 
Determining m

α

1
y
1
x
x

y


x
1
, y
1

and m form a
right
triangle


m

is the

hypotenuse


We can use the
Pythagorean

Theorem

to find m if we know
x
1

and y
1

m

2 2 2
1 1
2 2
1 1
m x y
m x y
 
 
perpendicular

perpendicular

1 box=1 cm

Moving Vectors

x

y


Just like arrows
vectors can be picked
up and moved
as
long as
we slide them
parallel

to the axes


No twisting allowed!

1 box=1 cm

Equivalent Vectors

x

y


Equivalent vectors
have


the same length


the same
orientation


the same
components

All these
vectors are
equivalent!!!

1 box=1 cm

Adding Vectors Graphically

x

y


Vectors can be
added graphically
by
placing them tail to tip


The result is a vector
drawn from the tail of
the first to the tip of
the last

A
B
C
A
1 box=1 cm

Adding Vectors Arithmetically

x

y


Vectors can be
added arithmetically
by adding their
components

A
B
C
C A B
C A B
x x x
y y y
 
 
A
x
C
x
B
x
A
y
C
y
B
y
1 box=1 cm

Adding Vectors


Order
doesn’t

matter

http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/vectors/ao.html

Sample Problems


Velocity of a plane with a head or tail wind


Boat crossing a river with a current


Net force on an object


Final displacement after a random walk


Negative Vectors

x

y


The
negative

of a
vector is a vector with
the
same magnitude
but the
opposite
orientation

A
A

1 box=1 cm

Subtracting Vectors

x

y


Reverse the vector
to be subtracted


Use the addition rule
to get the result

A
A B C
 
B
B

C
1 box=1 cm

Multiplying a Vector

by a Scalar

x

y


Multiply the
magnitude

of the
vector by the
scalar


The
orientation

of the vector
doesn’t change


OR


Multiply
each component
by
the
scalar
and construct the
new vector

B
A
4
B A

A
x
A
y
4
B A
y y

4
B A
x x

1 box=1 cm

Sample Problems


Newton’s Second Law:


Force equals mass times acceleration




F ma