Vectors

Physics 30S

How to Get from A to B?

• Task: Measure the distance from the door hinge to the supply room

• Materials:– Metre stick– Paper and pencil

How to Get from A to B – Part 2

• Task: Confirm your results by measuring again, without a ruler. Your second measurement should be within 5 cm. You may use the length of the tile is ____________.

• Materials:– Paper and pencil– Calculator

Is it enough?

• Travel your measured distance. • Are you back at the supply room?• Will that distance always get you to the supply

room?

What is a Vector?

• What is not a vector?• Most numbers so far have not been vectors;

they are scalars. – For example: 5, 7.5, ½, -13, π, etc

• Vectors are different because they have a direction!

Vector Notation

• On paper, vectors are signified with a half arrow above a capital letter

• In printed text, vectors are signified with a bolded capital letter– A

Vocabulary

• Scalar: magnitude only– Example: 4 out of 5, 23°C, 3

• Vectors: magnitude and a direction, with a unit– 3 tiles right, 5 metres left, 0.5 cm up

More about Vectors

• To specify a direction, we need a starting point, called a reference point

• Reference point: zero location in a coordinate system or frame of reference

• Position: the location of an object in relation to the reference point

What’s the Difference?

• Total distance travelled: sum total of actual steps taken; length of the path– scalar

• Displacement: shortest distance back to the start– vector

• Speed: how fast an object is moving– scalar

• Velocity: how fast an object is moving in a specified direction– vector

Homework

1. Write out the directions for how you got to school this morning. There should be enough detail for someone to follow the directions on a map!

2. Identify every vector in your writing in a list following your directions. Label these vectors D1, D2, D3, etc.

3. Vector Worksheet #1, 2

Significant Figures

• How long is the board?

Person Value measured for length1 11.6 cm2 11.6283476 cm3 11.63 cm

Is there a difference?

Does it Matter?

Recreate the net.Base of the square:7.6 cmHeight of the triangle:10.7 cm

Does it Matter?

Here are the actual measurements:

Base:7.6 (7.6200)Height:10.7 (10.6680)

Significant Figures

• Significant figures are an attempt to know how exact is a measurement– AKA. Sig figs

• For example, is the measurement 10.7, 10.67, 10.668, or 10.6680?

Definition

• Definition: Significant digits are those digits in a measurement that are known for certain plus one uncertain digit.

• When taking a measurement, record the last division plus estimate one more digit.

Practice Measurements

A)

B) Width of your pageC) Overhead items

Rules for Sig Figs1. All non zero digits are significant.

– 374 (3 sig figs)– 8.1 (2 sig figs)

2. All zeroes between other significant digits are significant.– 50407 (5)– 8.001 (4)

3. Leading zeroes in a decimal are not significant.– 0.54 (2)– 0.0098 (2)

4. Trailing zeroes are significant if they are to the right of a decimal point.– 2370 (3)– 16000 (2)– 160.0 (4)

5. Without a decimal, trailing zeroes are not significant.– 37000 (2)

What to Do About Zeroes?

In general:• If the zero is a placeholder, it is not significant.• If the zero does not need to be there, then it is

significant

Scientific Notation

• What if we know 5000 to 4 significant figures?• Use scientific notation:• 5.000 x 103

• Rule: Count the significant figures in the significand (leading number)

Practice Counting

A) 1174 km, N B) 5430 N, upC) 9.8 m/s2, down D) 0.006 N, downE) 3.00 x 108 m/s, right F) 909 cm, leftG) 6.0000 N, left H) 5060.050 μm,

right

Answers:A) 4 B) 3 C) 2 D) 1 E) 3 F) 3 G) 5 H) 7

Using Sig Figs in Calculations

• The least number of sig figs given is the number of sig figs that should be stated in the answer.

• Always round sig figs at the end of the question, not at each step!

Practice CalculationsA) 5.2 x 10.3 =B) 19.6 + 2.1 = C) 65 – 0.090 =D) 678.00 / 60 =E) (10.9 + 4) x 10.5 =

Answers:A) 54 B) 22 C) 65 D) 10 E)

200

Homework

• Pg.11 Glencoe Physics Study Guide• Sig Figs Worksheet– #4-14

Distance vs. Displacement

• Total distance travelled: sum total of actual steps taken; length of the path– Scalar

• Displacement: shortest distance back to the start– Vector – Displacement is the final position minus the initial

position

Drawing Vectors• Vectors are represented by an arrow• Length of the arrow = magnitude• Arrow points in the direction of the vector• Must be drawn to scale– Scale must be indicated

• Must draw a compass to indicate directions1 cm = 5 N

Directional Notation

• Degrees direction (N/S) of direction (E/W)– 25° S of E

• Direction (N/S) degrees direction (E/W)– S25 ° E

• Standard position angle

Multiplying Vectors by a Scalar

• Multiplying by a scalar multiplies the magnitude

• Multiplying by a negative reverses the direction

Examples

Drawa) Ab) 2Ac) –Ad) -3A

1 cm = 10 N

More Examples

Drawa) Ab) 1.5 Ac) -2.5 A

1 cm = 3 m/s

Homework

• Learning Activity 2.2– Pg. 15 Handout (Distance Ed)

Adding Vectors

Graphically (Tail to Tip method):• Draw one vector. • Draw the next vector at the tip of the first

vector. • Draw a new resultant vector from the

reference point to the end of the last vector• Measure the length and direction of the new

resultant vector

Example

A + B

1 cm = 5 m/s

A + B

Practice

Add these vectors using the tail to tip methoda) A + Bb) A – Cc) A + B + C

Adding Vectors Algebraically:1 Dimension

• Designate one direction as positive. All vectors going in this direction will be positive.

• The opposite direction will be negative. All vectors going in this direction will be negative.

• Sum the magnitude of the vectors together and interpret the direction!

Example

A + BLet E be positive.A is positive.B is negative.1 N - 2N = -1NA + B = -1 NA + B = 1 N, W

Homework

Add the following vectors using tail to tip method:1. A + B2. C + D

Add the following vectors algebraically:3. A + C4. B - D

Adding Vectors Algebraically:2 Dimensional Perpendicular

• Think back to tail to tip method

We can solve for W by using Pythagorean Theorem!

StepsStep 1: Draw a quick sketch.Step 2: Solve for the magnitude using

Pythagorean theorem.Step 3: Sketch in the resultant vector.Step 4: Solve for the direction using trigonometry.Step 5: Remember sig figs!

Example 1 A + B

Step 1: Draw a quick sketch.Step 2: Solve for the magnitude using

Pythagorean theorem.Step 3: Sketch in the resultant vector.Step 4: Solve for the direction using

trigonometry.Step 5: Remember sig figs!

Example 2

C + D

Example 3

C - D

Homework

• Assignment 2.1 (Distance Ed – P.45)• #1,2

Vectors Lab

Review

• A Vector Journey (Distance Ed. Pg. 49 -51)• Done with Sig figs• Sig fig practice

The Plan

• Max Classes: 81. What is a vector?2. Sig figs3. Drawing vectors/multiply by a scalar4. Adding vectors (tail to tip and algebraic in one

dimension)5. Adding vectors (2D)6. Vectors Lab and how to do a lab write up7. Review8. Test