Vectors and Velocity
Each point in space is specified by three numbers:
(
x
,
y
,
z
), and a
position vector
– an arrow showing
the displacement from the origin to that position.
Vectors add like successive displacements or
algebraically by
Vectors
Notation
We specify the directions we are
talking about by drawing
two little arrows of unit length
in two perpendicular directions.
“
x
”
and
“
y
”
are called
the coordinates
and can be positive
or negative.
Note that if
x
is negative, it means is a
vector pointing in the direction opposite to
Velocity
Average velocity is defined by
Instantaneous velocity is what we get
when we consider a very small time interval
(compared to times we care about)
Note: an average
velocity goes with
a
time interval
.
Note: an instantaneous
velocity goes with
a
specific time
.
We have 2 directions to specify. We must
–
Choose a reference point (origin)
–
Pick 2 perpendicular axes (
x
and
y
)
–
Choose a scale
We specify our
x
and
y
directions by drawing little
arrows of unit length in their positive direction.
A position vector (displacement from the origin)
is written
x
and
y
have units, do not.
Vectors (2-D coordinates)
1D Velocity
Velocity is the rate of change of position
Average velocity
= (how far did you go?)/(how long did it take you?)
Instantaneous velocity = same
(but for short Δ
t
)
9/9/15
6
Graphing velocity:
Figuring it out from the position
You can figure out the velocity
graph from the position graph using
Slope
= <
v
>
9/9/15
Position to velocity
Ratio of change in
position that takes
place to the (small)
time interval
Difference of two
positions at two
(close) times
We have 2 directions to specify. We must
–
Choose a reference point (origin)
–
Pick 2 perpendicular axes (
x
and
y
)
–
Choose a scale
We specify our
x
and
y
directions by drawing little
arrows of unit length in their positive direction.
A position vector (displacement from the origin)
is written
x
and
y
have units, do not.
Recall: Vectors (2-D coordinates)
Vector velocity and speed
A displacement – a change in position –
has a direction. This means
velocity = displacement/time interval
has one too.
We define speed as the magnitude of
velocity. (No vector on this. Why?)
9/9/15
Velocity to position
change in position that
takes place in
a small time interval
sum (
“
Σ
”
) in the
changes in position
over many small
time intervals
What have we learned?
Representations and consistency
Visualizing where
an object is
a position graph
at different times
Visualizing how fast
an object is moving
a velocity graph
at different times
Position graph
velocity graph
Velocity graph
position graph
Vectors
If a position vector (displacement
from the origin) lies in each of the
four indicated quadrants, what can
you say about the vector
representation of that
displacement,
1.
x
and
y
are both positive
2.
x
and
y
are both negative
3.
x
is positive and
y
is negative
4.
x
is negative and
y
is positive
x
y
A
B
C
D
Physics 131
9/9/15
Prof. E. F. Redish
Vectors in space specified by three numbers, position vectors, adding vectors algebraically, notation, velocity concepts, 2-D coordinates, 1D velocity, graphing velocity, position to velocity ratio. Explore how direction and displacement relate to velocity over time.
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Presentation Transcript
Vectors Each point in space is specified by three numbers: (x, y, z), and a position vector an arrow showing the displacement from the origin to that position. Vectors add like successive displacements or algebraically by A = Ax i + Ay j A+ B = Ax+ Bx B = Bx i + By j ) i + Ay+ By ( ) j (
Notation We specify the directions we are talking about by drawing two little arrows of unit length in two perpendicular directions. x and y are called the coordinates and can be positive or negative. Note that if x is negative, it means is a vector pointing in the direction opposite to r = x i + y j j i x i i
Velocity Average velocity is defined by v =Dr time it took to do it Note: an average velocity goes with a time interval. Dt=vector displacement Instantaneous velocity is what we get when we consider a very small time interval (compared to times we care about) r d v = Note: an instantaneous velocity goes with a specific time. dt
Vectors (2-D coordinates) We have 2 directions to specify. We must Choose a reference point (origin) Pick 2 perpendicular axes (x and y) Choose a scale We specify our x and y directions by drawing little arrows of unit length in their positive direction. A position vector (displacement from the origin) is written i , j i , x and y have units, do not. j
1D Velocity Velocity is the rate of change of position Average velocity = (how far did you go?)/(how long did it take you?) v =Dx Dt Instantaneous velocity = same (but for short t) v =dx dt
Graphing velocity: Figuring it out from the position Slope = <v> You can figure out the velocity graph from the position graph using v =Dx Dt x Dx = v Dt t Position as a function of time Velocity as a function of time 10 2 x (m) 8 1.5 6 x (m) 1 4 0.5 2 0 0 -5 0 5 10 15 -5 0 5 10 15 t (sec) t (sec) 9/9/15 6
Position to velocity x (m) v (m) dx dt v t (sec) t (sec) Ratio of change in position that takes place to the (small) time interval dx dt x t ( = v t ( ) + - D - D D ) x t ( ) t t Difference of two positions at two (close) times 2 2 = v t ( ) t 9/9/15
Recall: Vectors (2-D coordinates) We have 2 directions to specify. We must Choose a reference point (origin) Pick 2 perpendicular axes (x and y) Choose a scale We specify our x and y directions by drawing little arrows of unit length in their positive direction. A position vector (displacement from the origin) is written i , j i , x and y have units, do not. j
Vector velocity and speed A displacement a change in position has a direction. This means velocity = displacement/time interval has one too. dt vx i +vy j =d v =dr ( )= i + j dx dt dy dt x i + y j dt We define speed as the magnitude of velocity. (No vector on this. Why?) v = vx 2+vy 2
Velocity to position x (m) v (m) dx v t (sec) t (sec) dt change in position that takes place in a small time interval dt = ( ) dx v t dt sum ( ) in the changes in position over many small time intervals = = ( ) x dx v t dt 9/9/15
What have we learned? Representations and consistency Visualizing where an object is a position graph at different times Visualizing how fast an object is moving a velocity graph at different times Position graph velocity graph v =Dx v =dx slopes Dt dt areas Dx = v Dt Dx = vdt Velocity graph position graph
Vectors If a position vector (displacement from the origin) lies in each of the four indicated quadrants, what can you say about the vector representation of that displacement, 1. x and y are both positive 2. x and y are both negative 3. x is positive and y is negative 4. x is negative and y is positive y D A x C B
