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
1
2
3
Is there a difference?
Value measured for length
11.6 cm
11.6283476 cm
11.63 cm
Does it Matter?
Recreate the net.
Base of the square:
7.6 cm
Height 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 page
C) Overhead items
Rules for Sig Figs
1. 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
C) 9.8 m/s2, down
E) 3.00 x 108 m/s, right
G) 6.0000 N, left
B) 5430 N, up
D) 0.006 N, down
F) 909 cm, 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 Calculations
A)
B)
C)
D)
E)
5.2 x 10.3 =
19.6 + 2.1 =
65 – 0.090 =
678.00 / 60 =
(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 directions
1 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
Draw
a) A
b) 2A
c) –A
d) -3A
1 cm = 10 N
More Examples
1 cm = 3 m/s
Draw
a) A
b) 1.5 A
c) -2.5 A
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 method
a) A + B
b) A – C
c) 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+B
Let E be positive.
A is positive.
B is negative.
1 N - 2N = -1N
A + B = -1 N
A + B = 1 N, W
Homework
Add the following vectors using tail to tip method:
1. A + B
2. C + D
Add the following vectors algebraically:
3. A + C
4. B - D
Adding Vectors Algebraically:
2 Dimensional Perpendicular
• Think back to tail to tip method
We can solve for W by using
Pythagorean Theorem!
Steps
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 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: 8
1. What is a vector?
2. Sig figs
3. Drawing vectors/multiply by a scalar
4. Adding vectors (tail to tip and algebraic in one
dimension)
5. Adding vectors (2D)
6. Vectors Lab and how to do a lab write up
7. Review
8. Test