Kinematics & Dynamics in 2 & 3 Dimensions; Vectors

First, a reviewof some

Math Topicsin Ch. 1.Then, some

Physics Topicsin Ch. 4!

Vectors: Some Topics in Ch. 1, Section 7 General Discussion.

Vector A quantity with magnitude & direction.Scalar A quantity with magnitude only.

• Here, we’ll mainly deal with Displacement & Velocity. But, our discussion is valid for any vector!

• The Ch. 1 vector review has a lot of math! It requires a detailed knowledge of trigonometry!

Problem Solving• A diagram or sketch is helpful & vital!• I don’t see how it is possible to solve a

vector problem without a diagram!

Coordinate Systems Rectangular (Cartesian)

Coordinates• “Standard” coordinate axes.• A point in the plane is (x,y)• If its convenient, we could

reverse + & -

- ,+ +,+

- , - + , -

A “Standard Set” of xy Coordinate Axes

Vector & Scalar Quantities •Vector Quantity with magnitude & direction.•Scalar Quantity with magnitude only.

Equality of Two Vectors•Consider 2 vectors, A & B

A = B means A & B have the same magnitude & direction.

Vector Addition, Graphical Method • Addition of Scalars:

We use “Normal” arithmetic!• Addition of Vectors: Not so simple!• Vectors in the same direction:

– We can also use simple arithmetic• Example 1: Suppose we travel 8 km East on day

1 & 6 km East on day 2.Displacement = 8 km + 6 km = 14 km East

• Example 2: Suppose we travel 8 km East on day 1 & 6 km West on day 2.

Displacement = 8 km - 6 km = 2 km East “Resultant” = Displacement

Adding Vectors in the Same Direction:

Graphical Method of Vector Addition• For 2 vectors NOT along the gsame

line, adding is more complicated:• Example: D1 = 10 km East

D2 = 5 km North. What is the resultant (final) displacement?• 2 Methods of Vector Addition:

–Graphical (2 methods of this also!)–Analytical (TRIGONOMETRY)

Graphical Method of Adding Vectors“Recipe”

• Draw the first vector.• Draw the second vector starting

at the tip of the first vector• Continue to draw vectors “tip-to-tail”• The sum is drawn from the tail of the

first vector to the tip of the last vector

Example:

• Example: 2 vectors NOT along the same line. Figure!D1 = 10 km E, D2 = 5 km N.Resultant = DR = D1 + D2 = ?

• In this special case ONLY, D1 is perpendicular to D2.

• So, we can use the Pythagorean Theorem.

DR = 11.2 kmNote!

DR < D1 + D2

(scalar addition)

D1 = 10 km E, D2 = 5 km N.Resultant = DR = D1 + D2 = ?

The Graphical Method of Addition•Plot the vectors to scale, as in the figure.•Then measure DR & θ.Results in DR = 11.2 km, θ = 27º N of E

DR = 11.2 kmNote!

DR < D1 + D2

• This example illustrates general rules of graphical addition, which is also called the

“Tail to Tip” Method.• Consider R = A + B (See figure!).

Graphical Addition Recipe 1. Draw A & B to scale. 2. Place the tail of B at the tip of A 3. Draw an arrow from the tail of A to the tip of B

4. This arrow is the Resultant RMeasure its length & the angle with the x-axis.

Order Isn’t Important!Adding vectors in the opposite order gives the same

result: In the example in the figure,DR = D1 + D2 = D2 + D1

Graphical Method of Vector Addition

• Adding (3 or more) vectors:V = V1 + V2 + V3

• Even if the vectors are not at right angles, they can be added graphically using the tail-to-tip method.

• A 2nd Graphical Method of Adding Vectors (equivalent to the tail-to-tip method, of course!)

V = V1 + V2 1. Draw V1 & V2 to scale from a common origin.

2. Construct a parallelogram using V1 & V2 as2 of the 4 sides.3. Resultant V = Diagonal of the Parallelogram from a Common Origin

(measure length & the angle it makes with the x axis)

See Figure Next Page!

Parallelogram Method

Mathematically, we can move vectors around (preserving their magnitudes & directions)

A common error!

Parallelogram Method

Subtraction of Vectors • First, Define The Negative of a Vector:

- V vector with the same magnitude (size) as V but with opposite direction.

Math: V + (- V) 0 • Then add the negative vector.• For 2 vectors, V1 & V2:

Subtracting VectorsTo subtract one vector from another, add the first vector to

the negative of the 2nd vector, as in the figure below:

Multiplication by a Scalar• A vector V can be multiplied by a scalar c

V' = cV V' vector with magnitude cV & same direction as V.

• If c is negative, the resultant is in the opposite direction.

Example• Consider a 2 part car trip: • Displacement A = 20 km due North. • Displacement B = 35 km 60º West of North. • Find (graphically) resultant displacement vector R

(magnitude & direction). R = A + B. See figure below.

Use ruler & protractor tofind the length of R & theangle β.

Answers:Length = 48.2 km

β = 38.9º