Chapter 3: Two – Dimensional Motion and Vectors

Section 3-1 and 3-2

pages 84-97

Vectors

A scalar is a quantity that does not involve direction. 55 mph 18 cm long

A vector is a quantity that involves both magnitude and direction (velocity, acceleration, displacement, force) 55 mph north A downward force of 3 Newtons

Definition

Magnitude R is representedby length

Direction θ is representedby the angleTail

Head

θ

The resultant vector can be defined in polar coordinates as R at θ N of E.

Try YOURS!!

Parallel Vector Addition

10

13

Adding vectors in the same direction

=23 (resultant)

Adding vectors in the opposite direction

5

8

= 3 (resultant)

Basic Trig Functions

B = opp

A = adj

R= hyp

θ

x

y

For the right triangle placed at the origin

90o

Sin θ = B/R = opp/hypCos θ = A/R = adj/hypTan θ = B/A = opp/adjA2 + B2 = R2

Perpendicular Vector Addition

5

12

5

12

R

θR2 = 52 + 122

R = 13

θ = tan-1 5/12 = 22.6o

For two perpendicular vectors

Construct resultant R by drawing a vector from the tail of the horizontal vector to head of the vertical vector

Example:

8.00 m/s E

5.00 m/s N

A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s. What is the resultant velocity of the boat?

R = 9.43 m/s at 32°

Ө 1) Use pythagorean theory.2) Use tan Ө = opp/hyp

Multiple Vector Addition

AB

C

DE AB

C

D

E

R

A + B + C + D + E = DistanceR = Resultant = Displacement

R

Can be added in any order!!

Adding Vectors SUMMARYThe sum of two or more vectors is known as the

RESULTANT Vectors Acting in the Same Direction (parallel)

ADD 

 Vectors Acting in the Opposite Direction (parallel)

SUBTRACT

 At 90o angles – Ah- Trigonometry. . . (perpendicular)

PYTHAGOREAN

TAN Ө

At angles other than 90o - three methods

1. Graphical – scaled drawing

2. Resolution into Components Method – break each vector into right triangles then use trig functions

3. Law of Sines and Cosinescc22 = a = a22 + b + b22 – 2abcosC – 2abcosC

a a = = b b = = c c

sin A sin Bsin A sin B sin C sin C

Using the Graphical Method of Vector Addition:

Vectors are drawn to scale and the resultant is determined using a ruler and protractor.

Vectors are added by drawing the tail of the second vector at the head of the first (tip to tail method). The order of addition does not matter.

The resultant is always drawn from the tail of the first to the head of the last vector.

BE METICULOUS IN YOUR DRAWING!!! Your accuracy depends on it. (±2°, 0.2 cm)

Method 1: Adding Vectors Graphically (It’s making a scaled drawing.)

Steps:

Decide what quadrant the vectors will be in. Draw the axis and write the in a box.

 

Draw the first vector to scale starting at the origin and label it .

 

Draw the remaining vectors, so that they make a path and label them _____, ______, _____, etc.

 

Draw the as the dashed line from the and label it .

Measure the length of ____ to get the and the angle of

(from the closest axis) to get the and write your answer in a box.

DIRECTION ALWAYS < 45° angle DIRECTION ALWAYS < 45° angle of of ..

SCALE

A

B C D

RESULTANTTAIL OF 1ST VECTOR TO HEAD OF LAST

R

R MAGNITUDE

DIRECTION

R

TAIL TO HEAD

Example:

Cartman gets upset with Kenny for taking his doughnut. Cartman chases Kenny 30 meters at 40o N of E and then 20 meters at 10o E of N. Calculate Cartmans’s total displacement. Solve this graphically.

Example:

Cartman gets upset with Kenny for taking his doughnut. Cartman chases Kenny 30 meters at 40o N of E and then 20 meters at 10o E of N. Calculate Cartmans’s total displacement. Solve this graphically.

Advantages and Disadvantages of the Graphical Method

Can add any number of vectors at once

Uses simple tools No mathematical

equations needed

Must be correctly draw to scale and at appropriate angles

Subject to human error

Time consuming

This completes Method One!

So lets get

Vector problems #1 and #2 due tomorrow.