chap. 3: kinematics in two or three dimensions: vectors hw3: chap. 2: pb. 51, pb. 63, pb. 67; chap...

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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,

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