Download Scalar Quantity Vector Quantity

9/7/2011
VECTORS –
MEASURING FORCE AND
MOTION
The basics for measuring relationships of forces and
objects in physics
Scalar Quantity
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A quantity that can only be described in terms of
magnitude
Can be added, subtracted, multiplied and divided
like ordinaryy numbers
Examples include mass, time, temperature, volume,
distance, and speed
A scalar never mentions directions!
Vector Quantity
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Has both magnitude and direction for a complete
description
Can be added, subtracted, multiplied, and divided
like ordinaryy numbers
Examples include force, displacement, velocity, and
weight
A vector always mentions direction!
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Vector types we use in Physics
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Force – a push or a pull in a given direction
Displacement- How far an object moves, and in
what direction
Velocity – How fast something is moving and in
what direction
Representation of Vectors
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Represented graphically
by an arrowed line
The length of the line
g
indicates the magnitude
of the quantity
The direction of the
arrow indicates the
direction
50 m N
30 m E
VECTOR ADDITION
Calculating the net effect of two of more vectors
acting on the same object. (concurrently)
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Techniques for Vector Addition
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Place the vectors head to
tail
Draw an arrow from the tail
of the first vector to the
head of the second vector
Th sum off the
The
th vectors
t is
i
called the Resultant (R)
A+B=R
Equilibrant: The vector
that is equal in magnitude,
but in the opposite direction
of the resultant
B
A
3.0 m E
4.0 m E
R
7.0 m E
7.0 m W
Vector Addition Continued
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If vectors are heading in
the opposite direction
the same rules apply,
but one of the vectors
must be considered to
be “negative”
A + (-B) = R
B
3.0N E
A
1.5N W
R= 1.5N E
Adding Vectors at angles
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Vectors are not always acting in the same direction,
or directly opposing one another!
When vectors are at an angle to each other, the
pp y
laws of math must apply!
In physics, we will almost always be concerned with
vectors at right angles to each other
Therefore, right triangle mathematics will be
applied here!
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Try this
Joe walks 8 meters East then turns and walks 12
meters North.
14 m 56˚ N
of E
12 m N
8.0 m E
CONSTANT VELOCITY
(UNIFORM) MOTION
Moving in the same direction at the same speed
Some common phrases
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Motion – the change in position in time
Displacement – the distance and direction of an
object from a fixed reference point (a vector)
Average Velocity – The time rate of change of
displacement
Instantaneous velocity – velocity of an object at
any given point in it’s journey
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Uniform Motion
A closer look at the terms associated with
moving at a constant velocity
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Distance and direction
MUST considered
With Constant velocity
displacement is always
changing by the same
amount as time
i goes on
The graph of
displacement vs. time
would have a constant
slope!
The slope is equal to
average velocity
Displacement (m)
Displacement
Time (s)
rise Δy
slope =
=
run Δx
Average Velocity
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The time rate of change of displacement
Can be determined from the slope of a
displacement vs. time graph
Always Calculated over the entire time period you
want to look at!
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Equation for Average Velocity
v=
d
t
Where - v is average velocity
d is displacement
t is time
Sample Problem
A child is pushing a shopping cart at a speed of 1.5
m/s. How long will it take this child to push the cart
down an aisle with a length of 9.5 m?
v=d/t
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Given:
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v=1.5 m/s
d=9.5 m
9.5m
t
t = 9.5m / 1.5m / s
1.5m / s =
t = 6.33 3 s
t = 6.3s
**This is how every problem should be solved in physics to receive full credit.
The given information must be written down clearly. All equations that will be
used must be written down completely. And substitutions into the equations
must be shown.
Displacement vs. time graphs
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You can tell where the
object is relative to a fixed
position
You can determine the
(average) velocity of the
object from the slope of the
line
What is the displacement
over each section?
What is the average
velocity over each section?
Over the entire trip?
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C
3
Displacement (m)
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D
B
A
2
E
1
0
-1
0
1
2
3
4
5
6
7
10
Time (s)
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An Example for Average Velocity
Displacement (m) vs. Time
What is the displacement
between 0 and 10 sec?
50
What is the displacement
between 10 and 35 sec?
Displacement (m
m)
45
40
35
Total displacement?
30
25
20
What is the average
velocity for the first 10
sec?
15
10
5
0
-5
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Average for 10 s to 35
sec?
Time (s)
Another Example
Displacement vs. Time
50
A
45
Which object
has the greatest
average
velocity?
B
40
Displacement (m
m)
35
30
25
What is the
displacement of
object B?
20
15
10
5
0
0
1
2
3
4
5
6
7
8
Time (s)
Another Way to Find Average Velocity
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Don’t forget the simple
way to calculate an
average will also work
for velocity!
Final plus initial velocity
divided by 2 is
average velocity
V=
v f + vi
2
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Instantaneous Velocity
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Position (m)
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Time (s)
Velocity of an object at
any given point in its
journey
The slope
p of the tangent
g
line equals velocity at
that point!
This is the value we read
on the speedometer of
our cars
Practice problems
A squirrel descends an 8-m tree at a constant speed in 1.5
min. It remains still at the base of the tree for 2.3 min, and
then walks toward an acorn on the ground for .7 min. A
loud noise causes the squirrel to scamper back up the tree in
0.1 min to the exact position on the branch from which it
started. Which of the following
g graphs
g p would accuratelyy
represent the squirrel’s vertical displacement from the base
of the tree?
B
Time (min)
C
Time (min)
D
Position (m)
Position (m)
Position (m)
A
Position (m)
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Time (min)
Time (min)
Summing Up
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Displacement considers both magnitude and
direction
Average velocity can be determined from the slope
p
vs. time graph
g p
of a displacement
We will focus much more on average velocity than
instantaneous velocity
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