ntroduction: units, measurements, etc.
Review Problem:
A normal heartbeat rate is 60 beats/minute.
How many beats would you detect if you take someones
pulse for 10 sec instead of a minute?
Note:
magnitudes do not add unless vectors point in the
same direction Components of a Vector
Components of a Vector
A
A
component
component
is a part
is a part
It is useful to use
It is useful to use
rectangular components
rectangular components
These are the projections of the
These are the projections of the
vector along the
vector along the
x
x
-
-
and y
and y
-
-
axes
axes
Vector
Vector
A
A
is now a sum of its
is now a sum of its
components:
components:
y
x
A
A
r
r
r
+
=
A
What are and ?
x
A
y
A
r
r Components of a Vector
Components of a Vector
The
The
components
components
are the
are the
legs of the right triangle
legs of the right triangle
whose
whose
hypotenuse is
hypotenuse is
A
A
The
The
x
x
-
-
component
component
of a vector
of a vector
is the
is the
projection along the x
projection along the x
-
-
axis
axis
The
The
y
y
-
-
component
component
of a vector
of a vector
is the
is the
projection along the y
projection along the y
-
-
axis
axis
Then,
Then, cos
A
A
x
= sin
A
A
y
=
y
x
A
A
r
r
r
+
=
A
x
y
1
2
y
2
x
A
A
tan
and
A
A
A = +
=
y
A Notes About Components
Notes About Components
The previous equations are valid
The previous equations are valid
only if
only if
is
is
measured with respect to the x
measured with respect to the x
-
-
axis
axis
The
The
components
components
can be
can be
positive
positive
or
or
negative
negative
and
and
will have the same units as the original vector
will have the same units as the original vector What Components Are Good For:
What Components Are Good For:
Adding Vectors Algebraically
Adding Vectors Algebraically
Choose a coordinate system and sketch the
Choose a coordinate system and sketch the
vectors v
vectors v
1
1
, v
, v
2
2
,
,
Find
Find
the
the
x
x
-
-
and y
and y
-
-
components
components
of all the vectors
of all the vectors
Add
Add
all the x
all the x
-
-
components
components
This gives R
This gives R
x
x
:
:
Add
Add
all the y
all the y
-
-
components
components
This gives
This gives
R
R
y
y
: =
x
x
v
R
: =
y
y
v
R What Components Are Good For:
What Components Are Good For:
Adding Vectors Algebraically
Adding Vectors Algebraically
Choose a coordinate system and sketch the
Choose a coordinate system and sketch the
vectors v
vectors v
1
1
, v
, v
2
2
,
,
Find
Find
the
the
x
x
-
-
and y
and y
-
-
components
components
of all the vectors
of all the vectors
Add
Add
all the x
all the x
-
-
components
components
This gives R
This gives R
x
x
:
:
Add
Add
all the y
all the y
-
-
components
components
This gives
This gives
R
R
y
y
: =
x
x
v
R
Magnitudes of vectors
pointing
in the same
direction
can be added
to find the resultant!
: =
y
y
v
R Adding Vectors Algebraically
Adding Vectors Algebraically
Use the Pythagorean Theorem to find
Use the Pythagorean Theorem to find
the magnitude of the Resultant:
the magnitude of the Resultant:
Use the inverse tangent function to find
Use the inverse tangent function to find
the direction of R:
2
y
2
x
R
R
R
+
=
the direction of R:
x
y
1
R
R
tan =
IV. Motion in One Dimension
IV. Motion in One Dimension Dynamics
Dynamics
The branch of physics involving the
The branch of physics involving the
motion of an object and the relationship
motion of an object and the relationship
between that motion and other physics
between that motion and other physics
concepts
concepts
Kinematics
Kinematics
is a part of dynamics
is a part of dynamics
In kinematics, you are interested in the
In kinematics, you are interested in the
description
description
of motion
of motion
Not
Not
concerned with the cause of the
concerned with the cause of the
motion
motion Position and Displacement
Position and Displacement
Position
Position
is defined in terms
is defined in terms
of a
of a
frame of reference
frame of reference
Frame A:
Frame A:
x
x
i
i
>0
>0
and
and
x
x
f
f
>0
>0
Frame B:
Frame B:
x
x
i
i
<0
<0
but
but
x
x
f
f
>0
>0
One dimensional, so
One dimensional, so
generally the
generally the
x
x
-
-
or y
or y
-
-
axis
A
B
y
x
O
x
i x
f axis Position and Displacement
Position and Displacement
Position
Position
is defined in terms
is defined in terms
of a
of a
frame of reference
frame of reference
One dimensional, so
One dimensional, so
generally the
generally the
x
x
-
-
or y
or y
-
-
axis
axis
Displacement
Displacement
measures the
measures the
change in position
change in position
Represented as
Represented as
x
x
(if
(if
horizontal) or
horizontal) or
y
y
(if vertical)
(if vertical)
Vector quantity
Vector quantity
+ or
+ or
-
-
is generally sufficient to
is generally sufficient to
indicate direction for
indicate direction for
one
one
-
-
dimensional motion
Units
Units
Feet (ft)
Feet (ft)
US
US
Cust
Cust
Centimeters (cm)
Centimeters (cm)
CGS
CGS
Meters (m)
Meters (m)
SI
SI
dimensional motion Displacement
Displacement
Displacement
Displacement
measures
measures
the
the
change in position
change in position
represented as
represented as
x
x
or
or
y
y
m
m
m
x
x
x
i
f
70
10
80
1
+
= = = m
m
m
x
x
x
i
f
60
80
20
2 = = =
Distance or Displacement?
Distance or Displacement?
Distance may be, but is not necessarily, the
Distance may be, but is not necessarily, the
magnitude of the displacement
magnitude of the displacement
Displacement
(yellow line)
Distance
(blue line) Position
Position
-
-
time graphs
time graphs
Note:
position-time graph is
not necessarily a straight line
, even
though the motion is along x-direction ConcepTest
ConcepTest
1
1
An object (say, car) goes from one point in space
to another. After it arrives to its destination, its
displacement
is
1. either greater than or equal to
2. always greater than
3. always equal to
4. either smaller or equal to
5. either smaller or larger
than the
distance
it traveled.
Please fill your answer as
question 1
of
General Purpose Answer Sheet ConcepTest
ConcepTest
1
1
An object (say, car) goes from one point in space
to another. After it arrives to its destination, its
displacement
is
1. either greater than or equal to
2. always greater than
3. always equal to
4. either smaller or equal to
5. either smaller or larger
than the
distance
it traveled.
Please fill your answer as
question 2
of
General Purpose Answer Sheet ConcepTest
ConcepTest
1 (answer)
1 (answer)
An object (say, car) goes from one point in space
to another. After it arrives to its destination, its
displacement
is
1. either greater than or equal to
2. always greater than
3. always equal to
4. either smaller or equal to
5. either smaller or larger
than the
distance
it traveled.
Note: displacement is a vector from the final to initial points,
distance is total path traversed Average Velocity
Average Velocity
It takes time for an object to undergo a
It takes time for an object to undergo a
displacement
displacement
The
The
average velocity
average velocity
is
is
rate
rate
at which the
at which the
displacement occurs
displacement occurs
It is a
It is a
vector
vector
,
,
direction
direction
will be
will be
the same as
the same as
the
the
direction of the
direction of the
displacement
displacement
(
(
t
t
is always positive)
is always positive)
+ or
+ or
-
-
is sufficient for one
is sufficient for one
-
-
dimensional motion
t
x
x
t
x
v
i
f
average
=
=
r
r
r
r
dimensional motion More About Average Velocity
More About Average Velocity
Units of velocity:
Units of velocity:
Note:
Note:
other units may be given in a problem,
other units may be given in a problem,
but generally will need to be converted to these
Units
Units
Feet per second (ft/s)
Feet per second (ft/s)
US Customary
US Customary
Centimeters per second (cm/s)
Centimeters per second (cm/s)
CGS
CGS
Meters per second (m/s)
Meters per second (m/s)
SI
SI
but generally will need to be converted to these Example:
Example:
Suppose that in both cases truck
covers the distance in 10 seconds:
s
m
s
m
t
x
v
average
7
10
70
1
1
+
=
+
=
= r
r
s
m
s
m
t
x
v
average
6
10
60
2
2 = =
= r
r Speed
Speed
Speed is a
Speed is a
scalar
scalar
quantity
quantity
same units as velocity
same units as velocity
speed = total distance / total time
speed = total distance / total time
May be, but is not necessarily, the
May be, but is not necessarily, the
magnitude of the velocity
magnitude of the velocity Graphical Interpretation of Average Velocity
Graphical Interpretation of Average Velocity
Velocity can be determined from a position
Velocity can be determined from a position
-
-
time graph
time graph
Average velocity
Average velocity
equals the
equals the
slope
slope
of the line
of the line
joining the initial and final positions
joining the initial and final positions
s
m
s
m
t
x
v
average
13
0
.
3
40
+
=
+
=
= r
r Instantaneous Velocity
Instantaneous Velocity
Instantaneous velocity
Instantaneous velocity
is defined as the
is defined as the
limit
limit
of