Transcript Chapt3Class2

Chap. 3: Kinematics
in Two or Three
Dimensions: Vectors
HW3: Chap. 2: Pb. 51, Pb. 63, Pb. 67;
Chap 3:Pb.3,Pb.5, Pb.10, Pb.38, Pb.46
Due Wednesday, Sept. 16
Variable Acceleration; Integral
Calculus
Deriving the kinematic equations through
integration:
For constant acceleration,
Variable Acceleration; Integral
Calculus
Then:
For constant acceleration,
Displacement from
v av,x
Dx
= vx =
Dt
 Solve for displacement
Dx = v xDt
vx
vx
a graph of constant vx(t)
Dx
SIGN
+∆x
0
0
t1
t2
t
Displacement is the area between
the vx(t) curve and the time axis
-∆x
t
Displacement from graphs of v(t)
• What to do with a squiggly vx(t)?
Dx = v xDt
vx
o make ∆t so small that vx(t) does not change much
Displacement is the
area under vx(t) curve
Velocity does not
need to be constant
t1
∆t
t2
t
Graphical Analysis and Numerical
Integration
Similarly, the velocity may be written as the
area under the a-t curve.
However, if the velocity or acceleration is not
integrable, or is known only graphically,
numerical integration may be used instead.
One Dimensional Kinematics
https://www.youtube.com/watch?v=wNQzqCcTXR4&index=5&list=PLCFLie6gOOTx_CUIBUUXkhH2ezY8zcJB
Review Question
A ball is thrown straight up into the air.
Ignore air resistance. While the ball is in
the air the acceleration
A) increases
B) is zero
C) remains constant
D) decreases on the way up and increases on
the way down
E) changes direction
Vector Addition:
Graphical
Vectors
•
•
Scalars
Magnitude and
Direction
Magnitude only
Symbol
r
r
Examples:
Examples:
Displacement
Distance
Velocity
speed
acceleration
time
2D Vectors
•
How do I get to
Washington
from New
York?
•
Oh, it’s just
233 miles away.
Magnitude and direction are both required for a
vector!
Vector Addition: Graphical
• When we add vectors
Order doesn’t matter
A
A+ B = B+ A
B
We add vectors by drawing them “tip to tail ”
start
start
The resultant starts at the beginning of the first vector
and ends at the end of the second vector
Vector Addition Question
A
1)
Which graph shows the correct
placement of vectors for A + B
B
2)
3)
Vector Addition Question
A
1)
Which graph shows the correct
resultant for A + B
B
2)
3)
Vector Subtraction:
Graphical
When you subtract vectors, you add the vector’s opposite.
A - B = A+ - B
A
B
A+ B=C
-B
A
A-B= D
B
C
-B
A
A
D
Addition of Vectors—Graphical Methods
The parallelogram method may also be used; here
again the vectors must be tail-to-tip.
Multiplication of a Vector by a Scalar
A vector V can be multiplied by a
scalar c; the result is a vector c V
that has the same direction but a
magnitude cV. If c is negative, the
resultant vector points in the opposite
direction.
Vector Addition: Components
If the components are
perpendicular, they can be
found using trigonometric
functions.
Vector Addition: Components
• We don’t always carry around a ruler and a protractor,
and our result isn’t always very precise even when we
do. In this course we will use components to add
vectors.
• However, you should still always draw the vector
addition to help you visualize the situation.
• What are components here?
Parts of the vector that lie
on the coordinate axes
y
Ay = Asin q
A
q
Ax
Ay
x
Ax = Acosq
Vector Addition: Components
• We add vectors by adding their x and y components
because we can add things in a line
y
Ax = Acosq
y
A
q
x
Ay = Asin q
Ay
Ax
By
f
q
B
Bx = Bsin f
By = Bcos f
Bx
x
A
fB
Ax
Bx
Ay
By
C
C
Vector Addition:
Components
• We add vectors by adding their x and y components.
Ax
Bx
Ay
Cx = Ax + Bx
By
Cy = Ay + By
C
Ax
Bx
Cx
Cx
Ay
Cy
By
Cy
C
Vector Addition: Components
• Once we have the components of C, Cx and Cy, we
can find the magnitude and direction of C.
magnitude
Cx
a
C= C +C
2
x
2
y
Cy
C
direction
æ
ö
C
y
a = tan-1ç ÷
è Cx ø
South of East
Unit Vectors
Unit vectors have magnitude 1.
Using unit vectors, any vector V
can be written in terms of its
components:
Adding Vectors by
Components
Example 3-2: Mail carrier’s
displacement.
A rural mail carrier leaves the
post office and drives 22.0
km in a northerly direction.
She then drives in a direction
60.0° south of east for 47.0
km. What is her displacement
from the post office?
Vector Kinematics
In two or three
dimensions, the
displacement is a
vector:
Vector Kinematics
As Δt and Δr become
smaller and smaller, the
average velocity approaches
the instantaneous velocity.
Vector Kinematics
The instantaneous
acceleration is in the
direction ofΔ v = v2 – v1,
and is given by:
Vector Kinematics
Using unit vectors,