Chapter 2 One-Dimensional Kinematicsnsmn1.uh.edu/hpeng5/Peng03a_LectureOutline.pdfGiven that A + B = C, and that lAl2 + lBl2 = lCl2, how are vectors A and B oriented with respect to

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Chapter 2

One-Dimensional Kinematics

Description of motion in one dimension

Review:


Review: Motion with Constant Acceleration

Free fall: constant acceleration g = 9.81 m/s2


Chapter 3

Vectors in Physics


Units of Chapter 3• Scalars Versus Vectors

• The Components of a Vector

• Adding and Subtracting Vectors

• Unit Vectors

• Position, Displacement, Velocity, and Acceleration Vectors

• Relative Motion


3-1 Scalars Versus VectorsScalar: number with units

Vector: quantity with magnitude and direction

How to get to the library: need to know how far and which way

Vector: represented graphically by a line with an arrow

Length of the line: magnitudeArrow: direction


3-2 The Components of a VectorEven though you know how far and in which direction the library is, you may not be able to walk there in a straight line:


3-2 The Components of a VectorCan resolve vector into perpendicular components using a two-dimensional coordinate system:


3-2 The Components of a VectorLength, angle, and components can be calculated from each other using trigonometry:Ax = A cos θ, Ay = A sin θA: magnitudeOr:A = (Ax

2 + Ay2)1/2

θ = tan-1 (Ay / Ax)



Ax = A cos θ, Ay = A sin θA: magnitudeOr:A = (Ax

2 + Ay2)1/2

θ = tan-1 (Ay / Ax)

Equations used:

Ask: draw the vectors in blackboard?Express the vectors by unit vectors?


3-2 The Components of a Vector

Signs of vector components:


Hint: draw the vectors in x-y plane first!


Ask: 1. Express the vectors by unit vectors?2. Draw the vectors in blackboard?


Hint: draw the vector in x-y plane first!


Draw the vector in x-y plane in blackboard

Express the vectors by unit vectors


3-3 Adding and Subtracting VectorsAdding 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.



Adding vectors graphically

If two vectors are given such that A + B = 0, what

can you say about the magnitude and direction of vectors A and B?

a) same magnitude, but can be in any direction

b) same magnitude, but must be in the same direction

c) different magnitudes, but must be in the same direction

d) same magnitude, but must be in opposite directions

e) different magnitudes, but must be in opposite directions

Question 3.1aQuestion 3.1a Vectors IVectors I

If two vectors are given such that A + B = 0, what

can you say about the magnitude and direction of vectors A and B?

a) same magnitude, but can be in any direction

b) same magnitude, but must be in the same direction

c) different magnitudes, but must be in the same direction

d) same magnitude, but must be in opposite directions

e) different magnitudes, but must be in opposite directions

The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method.

Question 3.1aQuestion 3.1a Vectors IVectors I

Given that A + B = C, and that lAl 2 + lBl 2 = lCl 2, how are vectors A and Boriented with respect to each other?

a) they are perpendicular to each other

b) they are parallel and in the same direction

c) they are parallel but in the opposite direction

d) they are at 45° to each other

e) they can be at any angle to each other

Question 3.1bQuestion 3.1b Vectors IIVectors II

Given that A + B = C, and that lAl 2 + lBl 2 = lCl 2, how are vectors A and Boriented with respect to each other?






Note that the magnitudes of the vectors satisfy the Pythagorean Theorem. This suggests that they form a right triangle, with vector Cas the hypotenuse. Thus, A and B are the legs of the right triangle and are therefore perpendicular.

Question 3.1bQuestion 3.1b Vectors IIVectors II

Given that A + B = C, and that lAl + lBl = lCl , how are vectors A and B oriented with respect to each other?






Question 3.1c Question 3.1c Vectors IIIVectors III

Given that A + B = C, and that lAl + lBl = lCl , how are vectors A and B oriented with respect to each other?






The only time vector magnitudes will simply add together is when the direction does not have to be taken into account (i.e., the direction is the same for both vectors). In that case, there is no angle between them to worry about, so vectors A and B must be pointing in the same direction.

Question 3.1c Question 3.1c Vectors IIIVectors III


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.


3-3 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−

Q: components?


3-4 Unit VectorsUnit vectors are dimensionless vectors of unit length.


Express a vector in terms of unit vectors: yAxAA yx ˆˆ +=

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

that are equal in magnitude. Which of the that are equal in magnitude. Which of the

following is a possible angle for this vector following is a possible angle for this vector

in a standard in a standard xx--yy 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 components

that are equal in magnitude. Which of the that are equal in magnitude. Which of the

following is a possible angle for this vector following is a possible angle for this vector

in a standard in a standard xx--yy coordinate system?coordinate system?

a) 30°

b) 180°

c) 90°

d) 60°

e) 45°

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


Adding and Subtracting Vectors

Formula using unit vectors

yAxAA yx ˆˆ +=

yBxBB yx ˆˆ +=

yBAxBABA yyxx ˆ)(ˆ)( +++=+


3-4 Unit Vectors

Multiplying vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

yAxAA yx ˆ)(ˆ)( ααα +=

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 θ = 2y/2x, which is the same as tan θ = y/x (the original angle).

FollowFollow--upup:: if you double one component and not if you double one component and not the other, how would the angle change?the other, how would the angle change?

Question 3.2a Question 3.2a Vector Components IVector Components I

Summary of Chapter 3a

• Scalar: number, with appropriate units

• Vector: quantity with magnitude and direction

• Vector components: Ax = A cos θ, By = B sin θ

• Magnitude: A = (Ax2 + Ay

2)1/2

• Direction: θ = tan-1 (Ay / Ax)

• Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

• Component method: add components of individual vectors, then find magnitude and direction• Unit vectors are dimensionless and of unit length

Summary of Chapter 3


Documents

Chapter 2 One-Dimensional Kinematicsnsmn1.uh.edu/hpeng5/Peng03a_LectureOutline.pdfGiven that A + B = C, and that lAl2 + lBl2 = lCl2, how are vectors A and B oriented with respect to