Chap. 3: Chap. 3: Kinematics in Kinematics in Two or Three Two or Three Dimensions: Dimensions:
VectorsVectors
HW3: Chap. 2: Pb. 51, Pb. 63, Pb. 67; Chap 3:Pb.3,Pb.5, Pb.10, Pb.38, Pb.46Due 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 Displacement from
a graph of constant va graph of constant vxx(t)(t)
Solve for displacement
t1 t2
Displacement is the area between the vx(t) curve and the time axis
t
vx
+∆x
-∆x
SIGN
0 t v
x
0
Displacement from graphs of v(t)Displacement from graphs of v(t)• What to do with a squiggly vx(t)?
o make ∆t so small that vx(t) does not change much
t1 t2
Displacement is the area under vx(t) curve
t
vx
∆t
Velocity does not need to be constant
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 One Dimensional KinematicsKinematics
https://www.youtube.com/watch?v=wNQzqCcTXR4&index=5&list=PLCF-Lie6gOOTx_CUIBUUXkhH2ezY8zcJB
Review QuestionReview Question
A ball is thrown straight up into the air. Ignore air resistance. While the ball is in the air the accelerationA) increasesB) is zeroC) remains constantD) decreases on the way up and increases on the way downE) changes direction
Vector Addition: Vector Addition: GraphicalGraphical
Vectors Scalars
r r
Examples:Displacement
Velocity
acceleration
Distance
speed
time
Examples:
2D Vectors2D Vectors
Magnitude and direction are both required for a vector!
• How do I get to Washington from New York?
• Oh, it’s just 233 miles away.
Vector Addition: GraphicalVector Addition: Graphical
• When we add vectors
Order doesn’t matter
We add vectors by drawing them “tip to tail ”
A B
start start
The resultant starts at the beginning of the first vectorand ends at the end of the second vector
Vector Addition QuestionVector Addition Question
A B
1) 2) 3)
Which graph shows the correct placement of vectors for +
A B
Vector Addition Vector Addition QuestionQuestion
A B
1) 2) 3)
Which graph shows the correct resultant for +A B
Vector Subtraction: Vector Subtraction: GraphicalGraphical
A B
When you subtract vectors, you add the vector’s opposite. - = + -
A B A B
B-A
CDA
B -B
A
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 can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.
V
V
Vector Addition: Components
If the components are perpendicular, they can be found using trigonometric functions.
Vector Addition: Vector Addition: ComponentsComponents
• 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?
A
x
y
Ax
Ay
Parts of the vector that lie on the coordinate axes
Vector Addition: Vector Addition: ComponentsComponents
• We add vectors by adding their x and y components because we can add things in a line
A
x
y
Ax
Ay
B
y
x
Bx
By
C C
Ax
By
Bx
Ay
A
B
Vector Addition: Vector Addition: ComponentsComponents
• We add vectors by adding their x and y components.
C
Ax
By
Bx
Ay
Ax Bx
Cx
By
Ay
Cy
Cx
CCy
Vector Addition: Vector Addition: ComponentsComponents
• Once we have the components of C, Cx and Cy, we can find the magnitude and direction of C.
C
Cx
Cy
South of East
magnitude
direction
Unit VectorsUnit Vectors
V
Unit vectors have magnitude 1.
Using unit vectors, any vector 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 KinematicsVector Kinematics
In two or three dimensions, the displacement is a vector:
Vector KinematicsVector Kinematics
As Δt and Δr become smaller and smaller, the average velocity approaches the instantaneous velocity.
Vector KinematicsVector Kinematics
v
v
v
The instantaneous acceleration is in the direction ofΔ = 2 – 1, and is given by:
Vector KinematicsVector KinematicsUsing unit vectors,
