Vectors

Bellwork ( Clickers )

Some Notes

Stations ( Take Notes!! )

Open Notes/Book Quiz on Thur ( 09/13/12 )

What’s On Quiz – 6 Qs

• Conceptual Questions on Vectors – Conceptual Clickers / Stations

• Finding resultants/angles of vectors (free-response)– Graphical and Analytical– AP will have 3 of these ( Honors only 2 )

If each component of a

vector is doubled, what

happens to the angle of

that vector?

a) it doubles

b) it increases, but by less than double

c) it does not change

d) it is reduced by half

e) it decreases, but not as much as half

Question 3.2a Question 3.2a Vector Components IVector Components I

If each component of a

vector is doubled, what

happens to the angle of

that vector?

a) it doubles

b) it increases, but by less than double

c) it does not change

d) it is reduced by half

e) it decreases, but not as much as half

The magnitude of the vector clearly doubles if each of its components is doubled. But the angle of the vector is given by tan = 2y/2x, which is the same as tan = y/x (the original angle).

Follow-upFollow-up:: if you double one component and not the if you double one component and not the other, how would the angle change?other, how would the angle change?

Question 3.2a Question 3.2a Vector Components IVector Components I

A certain vector has A certain vector has xx and and yy components that are components that are

equal in magnitude. Which of the following is a equal in magnitude. Which of the following is a

possible angle for this vector in a standard possible angle for this vector in a standard x-yx-y

coordinate system?coordinate system?

a) 30°

b) 180°

c) 90°

d) 60°

e) 45°

Question 3.2b Question 3.2b Vector Components IIVector Components II

A certain vector has A certain vector has xx and and yy components that are components that are

equal in magnitude. Which of the following is a equal in magnitude. Which of the following is a

possible angle for this vector in a standard possible angle for this vector in a standard x-yx-y

coordinate system?coordinate system?

a) 30°

b) 180°

c) 90°

d) 60°

e) 45°

The angle of the vector is given by tan = y/x. Thus, tan = 1 in this

case if x and y are equal, which means that the angle must be 45°.

Question 3.2b Question 3.2b Vector Components IIVector Components II

The Components of a Vector**Notice arrow above and bold font / Angle goes counterclockwise!!

Can resolve vector into perpendicular components using a two-dimensional coordinate system:

The Components of a VectorLength, angle, and components can be calculated from each other using trigonometry:

The Components of a Vector

Signs of vector components:

Adding and Subtracting Vectors

Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last.

Finding Vector Resultants

• Graphical method is approximate– Limited by accuracy of graph, measurements, etc.

• Using components ( analytical ) is more accurate

Outdoor Lab Example

Building

Light-pole

Tree

Components

Resultant ( Sum )

Adding and Subtracting Vectors

Adding Vectors Using Components:

1. Find the components of each vector to be added.

2. Add the x- and y-components separately.

3. Find the resultant vector.

Adding and Subtracting Vectors

Adding and Subtracting Vectors

Subtracting Vectors: The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, .

D= A B

Examples

• Ax = (5.00 m)*cos 60.0o Ay = (5.00 m)*sin 60.0o

• Bx = (4.00 m )*cos 20.0o By = (4.00 m )*sin 20.0o

• Add x and y components separately• Cx is 6.26 m and Cy is 5.70 m

Adding and Subtracting Vectors

Examples – Take a Step Further

Examples – Take a Step Further

Stations – Take Notes!!• #1 – Basic conceptual / 1D Kinematics review

– Average speed ( distance / time )– Average velocity ( displacement / time or:

• #2 – Basic Resolution of Vector into Components

• #3 – Vector Addition ( starts empty )

• #4 – PhET Vector Addition Simulation

• #5 – Playing with Force Table at front of room

Question 3.3Question 3.3 Vector AdditionVector Addition

You are adding vectors of length 20

and 40 units. What is the only

possible resultant magnitude that

you can obtain out of the following

choices?

a) 0a) 0

b) 18b) 18

c) 37c) 37

d) 64d) 64

e) 100e) 100

Question 3.3Question 3.3 Vector AdditionVector Addition

You are adding vectors of length 20

and 40 units. What is the only

possible resultant magnitude that

you can obtain out of the following

choices?

a) 0a) 0

b) 18b) 18

c) 37c) 37

d) 64d) 64

e) 100e) 100

The minimumminimum resultant occurs when the vectors are

oppositeopposite, giving 20 units20 units. The maximummaximum resultant

occurs when the vectors are alignedaligned, giving 60 units60 units.

Anything in between is also possible for angles

between 0° and 180°.