3

Table of Contents:

1. Significant Figures (pages 9-11 in the book) Page 4

2. Scientific Notation Page 5

3. The Metric/SI System Page 6

4. Factor Label Unit Conversion Page 7

a. Units and Dimensional Analysis Homework Page 8

5. Unit Analysis Page 9

6. Solving for Variables Page 10 – 11 a. Solving for Variables Homework Page 12

7. Graphing Page(s) 13 – 14 a. The Cartesian Coordinate system Page 15

8. Trigonometry Page 16 a. Introduction to Trigonometry Page 16 b. Inverse Trigonometry Page 17 – 18 c. Trigonometry homework Page 19 – 20

9. Vectors Page(s) 21 – 28 a. Vector Components Page(s) 22 – 23 b. Vector practice Page 24 c. Vector Addition Page 25 d. Adding Vectors Mathematically Page 26 e. Adding Vectors Graphically Page 27 f. Vectors homework Page 28

4

Significant Figures

All measurements have uncertainties, we often indicate the precision of a number by writing it with the symbol “±” followed by a number indicating the likely uncertainty. In AP Physics we will keep everything to 3 significant figures.

Rules for deciding which digits are significant: 1. Nonzero digits are always significant. 2. All final zeros after the decimal point are significant. 3. Zeros between two other significant digits are always significant. 4. Zeros used solely for spacing the decimal point are not significant.

Examples: Using the rules above, 0.0340 mm has 3, 960 kg has 2, and 70,080 s has 4 significant figures. In addition or subtraction, work with the numbers as they are written, but the final answer must have the same level of precision as the least precise number – in the case below, precision to the tenths place.

123.479 m + 35.8 m + 5.32 m = 164.599 m = 164.6 m

In multiplication or division, work with the numbers as they are written, but the final answer must have the same number of significant digits as the number with the least amount of digits – in the case below, 3 significant digits.

𝟏𝟐𝟑. 𝟒𝟕𝟗 𝑲𝒈 ÷ 𝟑𝟓. 𝟖 𝒎𝟑 = 𝟑. 𝟒𝟒𝟗𝟏𝟑𝟒𝟎𝟕𝟖 = 𝟑. 𝟒𝟓

In multiplication or division, units are multiplied and divided like numbers.

𝟕.𝟒𝟖 𝒎𝟑

𝟑.𝟔 𝒎 = 2.1 m2

Exercises 1. State the number of significant digits in each of the following measurements.

a. 32.06 kg ________ b. 0.02 km ________ c. 5400 m ________ d. 2006 s ________

e. 2.9910 m ________ f. 5600 km ________ g. 0.00670 kg ________ h. 809 g ________

2. Solve the following problems and give the answers to the correct number of significant digits. (All numbers must be written with the correct units.)

a. 324.54 cm - 25.6 cm = ________________ b. 28.9 g + 300.25 g + 2.945 g = ________________

c. 82.3 m x 1.254 m = ___________________ d. (1.2 x 106 m)(3.25 x 104 m) = _________________

e. 𝟑𝟐.𝟔 𝒌𝒈

𝟏𝟐𝟓.𝟒 𝑳 = ____________________________ f.

𝟒.𝟐𝟒 ×𝟏𝟎−𝟑 𝒌𝒈

𝟐.𝟐 ×𝟏𝟎−𝟒 𝑳 = _____________________________

g. 𝟐.𝟔𝟒𝟕 𝒎

𝟐.𝟎 𝒎 = __________________________ h. 5.25 cm x 1.3 cm = _________________________

i. 9.0 cm + 7.66 cm + 5.44 cm = ___________ j. 10.07 g - 3.1 g = ____________________________

5

Scientific Notation

Scientific notation is a compact way of writing large or small numbers while using only significant digits and powers of ten. The normal form for scientific notation is:

A x 10 b a is a number in the form of 0.00 (i.e. a digit in the ones, tenths, and hundredths place) b is the powers of ten telling us which direction to move the decimal place and how many times.

Examples: 0.00265 written in scientific notation would be 2.65 x 10-3 (The negative three power of ten indicates that the decimal point should be moved three places to the left.)

7.68 x 105 expanded would be 768,000 (The positive power of five indicates that the decimal point should be moved five places to the right. In this case, zeros are needed as placeholders.)

Divide the decimals while keeping the correct number of significant figures. When dividing powers of ten, subtract the bottom power of ten’s exponent from the top power of ten’s exponent. If multiplying, add powers of ten.

𝟗.𝟔 × 𝟏𝟎𝟕

𝟑.𝟐 × 𝟏𝟎𝟒 = 𝟑. 𝟎 𝐱 𝟏𝟎𝟑.

Before adding or subtracting numbers in scientific notation, all numbers need to have the same power of ten.

(4.5 x 10-2) + (8.2 x 10-3) = 4.5 x 10-2 + 0.82 x 10-2 = 5.32 x 10-2 = 5.3 x 10-2

Exercises: Write the following numbers in scientific notation 1) 5800 m _______________ 2) 0.0345 mm_______________ 3) 0.00890 kg _______________

4) 560 g ________________ 5) 43 200 L _______________ 6) 4 320 000 cm2_____________

7) 0.00065 m/s_____________ 8) 101.35 m_______________ 9) 2.004 sec_________________

Expand the following numbers: 10) 1.20 x 103 km_______________ 11) 2.34 x 10 ─ 4 m___________ 12) 2.00 x 100 m3_____________

13) 1.99 x 104 kg_____________ 14) 9.80 x 10─1 L____________ 15) 6.65 x 10─5 mm ____________

6

Le Système International d'Unités (SI System of measurement) There are two quantities required of every measurement:

1. Dimension – The physical quantity being measured, such as length, mass, or force. The dimension of a measurement is indicated by the unit following the number.

2. Unit – This indicates the size of the measurement that is made, for example a mile is larger than a foot which is larger than inch.

The Metric System differs mainly from the British system of measurement in the way it indicated the size of a measurement. In the British system of measurement different units are used to measure larger amounts. When an inch becomes too small a foot is used instead, in the metric system of measurement instead of changing the units, prefixes are applied to base units (each of which only represents a single dimension) to indicate larger or smaller measurements. The SI system is a component of the metric system, the only difference is that the SI system uses certain standard units for dimensions of measurement. This is meant to better standardize the measurements throughout different sciences; these are referred to as SI units and are used to make all other units of measurement in different combinations. The SI UNITS are as follows:

Length – metres (m)

Mass – kilograms (kg)

Time – seconds (s)

Ampere – electric current (A)

Kelvin – temperature (K)

Candela – luminous intensity (cd)

Mole – The amount of substance. (mol) Whenever any mathematical calculations are performed the units of all quantities involved must be in SI units.

All other units in the metric system are different combinations of SI units, we call these units derived units. The metric prefixes are displayed below and the common prefixes are bolded.

Order of Magnitude Prefix Abbreviation

106 Mega - M

103 Kilo - K

102 Hecto - H

101 Deka - da

100 BASE ------

10 -1 deci - d

10-2 centi - c

10-3 milli - m

10-6 micro - 𝜇

10-9 nano - n

7

Factor-Label Unit Conversion

For the problems below, show all conversion factors, work. And do not use the conversion factor feature on your calculator to do these problems.

1. Carry out the following conversions using the prefix information shown above.

a. 35 nm = ___________𝑚 b. 450 cm = ___________𝑚𝑚

c. 1500 µg = ___________𝑚𝑚 d. 250 km = ___________𝑐𝑚

e. 346 ms = ___________𝑠 f. 543 mg = ___________𝑘𝑔

g. 4008 g = ___________𝑚𝑔 h. 239 mm = ___________𝑐𝑚

i. 48 mL = ___________𝐿 j. 38 kg = ___________𝑚𝑔

Common Prefixes: nano has the symbol, n, and means 10–9 (or 0.000000001)

micro has the symbol, µ , and means 10 –6 (or 0.000001)

milli has the symbol, m, and means 10–3 (or 0.001)

centi has the symbol, c, and means 10–2 (or 0.01)

kilo has the symbol, k, and means 103 (or 1000)

8

1. Compute the number of seconds in an hour, day, and year.

2. The maximum sodium intake for a person on a 2000 calorie diet should be 2400 mg/day. How many grams

of sodium is this per day?

3. The recommended daily allowance (RDA) of the trace element chromium is 120 μg/day. Express this dose

in grams per day

4. An intake of vitamin C up to 500 mg/day is considered safe. Express this intake in grams per day.

5. The electrical resistance of the human body is approximately 1500 ohms when it is dry. Express this

resistance in Kilohms.

6. An electrical circuit of about 0.020 amps can cause muscular spasms so that a person cannot, for example

let go of a wire. Express this current in milliamps.

7. Starting with the definition 1.00 inches = 2.54cm, find the number of kilometers in 1.0 miles.

8. In medicine, volumes are often expressed in milliliters; prove that a milliliter is the same as a cubic

centimeter.

9. How many cubic centimeters are there in a 1.00 liter bottle of drinking water?

(Hint: 1 cm3 = 1 mL)

10. Bacteria vary somewhat in size, but a diameter of 2.0μm is not unusual. What would be the volume (in

cubic centimeters) and the surface area (in square millimeters) of such bacterium assuming that it is

spherical? (Consult Appendix A of your book for assistance)

Units and Dimensional Analysis

9

Unit Analysis

Unit analysis involves the placement of units in an equation to check for unit agreement on both sides of the equals sign. When substituting values into an equation in physics, you must state the units as well as the numerical values. Including units in your calculations helps you keep units consistent throughout and assures you that your answer will be correct in terms of units. The proper units for variables are included in the table below.

Examples

Equation with Unit Agreement

The equation being inspected is �̅� = 𝑑

𝑡 its units

indicate that 𝑚

𝑠=

𝑚

𝑠

The units on the left are equal to the units on the right of the equal sign. This equation is correct in terms of units.

Equation with Unit Disagreement

The equation being inspected is 𝑣𝑓 = 𝑣𝑜2 + 2𝑎𝑑 its units indicate that

𝑚

𝑠= (

𝑚

𝑠)

2+ (

𝑚

𝑠2) 𝑚2

(Notice that coefficients, such as the number 2, in the equation are ignored in dimensional analysis)

The units on the left are 𝑚

𝑠 but the units on the right turn out to be

𝑚2

𝑠2+

𝑚3

𝑠2

The units do not agree. This final velocity equation is incorrect in terms of its units.

Problems Use the method described above to determine if the following equations have unit agreement. Show your work and write “correct” next to the equation if it is correct and “incorrect” if it is incorrect in terms of its units.

1. 𝑣2 = 𝑣1 + 𝑎𝑡 2. �̅� = 12⁄ (𝑣𝑜 + 𝑣𝑓) 3. 𝑑 = 𝑣1𝑡 +

1

2𝑎𝑡

4. 𝑣22 = 𝑣1

2 + 2𝑎𝑑 5. 𝑑 = √𝑣1𝑡2 + 1

2𝑎𝑡

Quantities Units Units

(d) displacement (m) Meters

(vo) original velocity (vf) final velocity

(�̅� ) average velocity

(meters per second) m/s

(a) acceleration (meters per second

squared) m/s2

(t) time s (seconds)

10

Solving for Variables PROBLEM SOLVING When solving motion problems, or, indeed, any physics problem, use an orderly procedure like the one listed below. 1. Identify the quantities that are given in the problem. 2. Identify the quantity that is unknown, the one you have to find. 3. Select the equation that contains the given and unknown quantities. 4. Solve the equation for the unknown quantity using algebra. 5. Substitute the values given in the problem, along with their proper units, into the equation and solve it. 6. Check to see if the numerical value of your answer is reasonable and make sure that the answer has the correct units.

Example Solve for each variable in the equation below. The variable should be placed alone on the left side of the equation.

𝑣2 = 𝑣1 + 𝑎𝑡 (It is already solved for v2, so it only needs to be solved for v1, a, and t.)

𝑣1 = 𝑣2 − 𝑎𝑡 𝑣2− 𝑣1

𝑡= 𝑎 𝑡 =

𝑣2− 𝑣1

𝑎

Problems 1. In the equation, 𝑑 = 1

2⁄ (𝑣1 + 𝑣2)𝑡, solve for v2 and t.

2. In the equation, 𝑑2 − 𝑑1 = 𝑣𝑜𝑡 + 1

2𝑎𝑡2 , solve for vo and a.

3. In the equation, 𝑣22 = 𝑣1

2 + 2𝑎𝑑, solve for v2, v1, and d.

11

Solving for Variables Examples

Solve the following equation for c.

𝐸 = 𝑚𝑐2

(Divide both sides by m)

𝐸

𝑚= 𝑐2

(Take the square root of both sides and there will be a positive and negative solution for c)

±√𝐸

𝑚= 𝑐 or finally, 𝑐 = ±√

𝐸

𝑚

Solve the following equation for k.

3 √𝑘

𝑔= 𝑑

(Multiply both sides by g)

3 √𝑘 = 𝑔 𝑑 (Divide both sides by 3)

√𝑘 =𝑔 𝑑

3

(square both sides of the equation)

𝑘 =𝑔2 𝑑2

9

Problems Solve the following equations for the variable(s) requested.

1. E = ½mv2 for m and v 2. 𝑃 = 𝐹𝑑

𝑡 for d

3. E = mgh for h 4. E = hf - Wo for Wo and h

5. 𝑠𝑜

𝑠𝑖=

𝑑𝑜

𝑑𝑖= for si 6. ax2 + b = c for x

Solve for x in the following problems.

7. 3𝑥

𝑦=

6𝑔

𝑏___________________________ 8.

2𝑥2

3= 𝑑𝑔 ___________________

9. 𝑡

𝑥= 𝑑 ___________________________ 10.

2√𝑥

𝑐= 𝑦 ___________________

12

Solving for Variables Directions: Solve for the following for the variable a please show all of your work in a step by

step method, and circle your answers.

1. 𝑎 + 𝑏 = 2𝑎

2. 𝑏 + √𝑏𝑎 = 1

3. √𝑎 + 𝑏 = 𝑐𝑥

4. 1

𝑎−

1

𝑏=

1

𝑐

5. 𝑎2𝑧 + 1

𝑥=

1

𝑧

13

Graphing

Plotting Graphs These steps will help you plot graphs from data tables. 1. Your graph should be titled Y vs. X (or the vertical information versus the horizontal information). 2. Decide on the scales needed for the x and y axes. Choose scales that will spread out the data.

Do not choose scales that compress the data points into a tiny portion of the graph paper. Your graph should fill up the graph paper.

3. Number and label the x and y axes (including necessary units).

4. Draw the best straight line (using a straight edge) or smooth curve that passes through as

many data points as possible. Do not just connect the data points together with a series of straight line segments.

Graphing Data Values The steps listed above were followed to set up the plotting of the data shown below.

Time (s)

Speed (m/s)

0 4

1 15

2 20

3 37

4 55

5 59

Exercises 1. Plot the data values on the graph provided above and draw one straight line that best fits the data (use a straight edge to make the line). 2. What is the slope of the line of best fit (find the number)? _____________________ m/s2

(Slope = rise/run) 3. What is the speed at 3 s? ________________________________________________ m/s 4. Using the graph, what is the speed at 6 s? __________________________________ m/s 5. At what time is the speed 20 m/s (using your graph) ___________________________ s

14

Graphing There are three relationships that occur frequently in physical processes. They are depicted in the three graphs shown below.

Exercises 1. Plot the data values on the graph provided above and connect the points together with a smooth curve that follows the data. 2. Approximately what is the radius when the speed is 5 m/s? _______________________m. 3. This type of curve is known as a _______________________________________________. 4. This graph follows an equation of the form radius = k(speed)2. Radius and speed squared are ________________________________ related.

Speed (m/s)

Radius (m)

0 0

1 4 2 16

3 36 4 48

15

+𝜃

−𝜃

The Cartesian coordinate system

The Cartesian coordinate system is a linear coordinate system that we will be using most often when creating graphs and plots, or when describing the directions of a vector. This system should be familiar to you, from math class as the x – y coordinate system.

Angles and reference points: When measuring angles, we must do so from either the ±x or ± y axis.

When an angle is measured from an axis in a counter clockwise rotational direction its value is positive.

When an angle is measured from an axis in a clockwise rotational direction its value is negative.

An important point to be aware of is that whenever measuring an angle we ALWAYS use the positive x – axis our reference axis. Therefore the angle made with the positive x – axis is referred to as the reference angle.

When measuring an angle from the positive x – axis your calculator will know exactly where that specific angle is.

When measuring an angle from the negative x – axis your calculator will automatically assume the angle is in the first quadrant (So do not allow your calculator to make a fool out of you.)

Make sure you know your positive directions from your negative directions. There are four Quadrants:

I. From 0o to 90o x = positive values, y = positive values II. From 90o to 180o x = negative values, y = positive values

III. From 180o to 270o x = negative values, y = negative values IV. From 270o to 360o or 0o x = positive values, y = negative values

Draw the following approximate angles.

I. 45o

II. 150o

III. 110o

IV. 335o

V. 240o

0o 180o

90o

270o

16

Trigonometry Trigonometry is an extremely useful branch of mathematics that deals with the relationships between the sides and angles of right triangles.

The opposite side is the side which the angle does not touch. The adjacent side is the side of the triangle that is not the hypotenuse but is part of the angle itself.

It is to be noted that these two sides can change as you

use different reference angles.

Observe Figure 1 at the top of the page Using 𝜽 as our reference angle the sides are as follows:

Side a = the opposite side Side b = the adjacent side

Using 𝜷 as our reference angle the sides are as follows:

Side a = the adjacent side Side b = the opposite side

The Pythagorean Theorem is a

method for finding missing sides of right

triangles.

It states that the hypotenuse (the longest side) is

related to the other two sides of a triangle by

the following equation.

𝑎2 + 𝑏2 = 𝑐2

(where c is the length of the hypotenuse and a and b are

the lengths of the other two sides)

The three commonly used trigonometric functions

are (using 𝜃 as our reference angle)

sine: 𝒔𝒊𝒏(𝜽) = 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆

𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆=

𝒂

𝒄

Cosine: 𝑪𝒐𝒔 (𝜽) = 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆

𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆=

𝒃

𝒄

Tangent 𝒕𝒂𝒏 (𝜽) = 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆

𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆=

𝒂

𝒃

(Memorizing SOH CAH TOA is one way of

memorizing these trigonometric functions.)

1. For triangle I above, cos(A) = _____

2. For triangle I above, tan(A) = _____

3. For triangle II, c = _____

4. For triangle III b = _____

𝜃

𝛽

a

b

c

Figure 1

A =22.6o

Using ∠𝐴 (13) cos(22.6o) = 12 (13) sin(22.6o) = 5

Sides may also be found using trig functions and solving for specific sides.

Using ∠𝐵

(13) cos(67.4o) = 5 (13) sin(67.4o) = 12

17

Inverse Trigonometry Functions

We learned that we use one of the trig functions, to take an angle of a triangle and find the side length. When using “special” trig functions we will do the opposite, take the side lengths and find the angle. Because this is the opposite operation, we call these functions inverse functions of each of the trig ratios we saw before. The notation we will use for the inverse trig functions will be similar to the inverse notation we used with functions.

sin−1 (𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) = 𝜃 cos−1 (

𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) = 𝜃 tan−1 (

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡) = 𝜃

Just as with inverse functions, the − 1 is not an exponent, it is a notation to tell us that these are inverse functions. While the regular trig functions take angles as inputs, these inverse functions will always take a ratio of sides as inputs. We can calculate inverse trig values using a table or a calculator (usually pressing shift or 2nd first). Since in AP Physics we will only be dealing in two dimensions, and will ALWAYS use a reference angle, relative to the positive x axis we will only need to use the inverse tangent function.

Example: sin θ = 0.5 We don′t know the angle so we use an inverse trig function

sin−1(0.5) = θ Evaluate using a calculator θ = 30o Our Solution

cos θ = 0.667 We don′t know the angle so we use an inverse trig function

cos−1(0.667) = θ Evaluate using a calculator θ = 48o Our Solution

tan θ = 1.54 We don′t know the angle so we use an inverse trig function tan−1(1.54) = θ Evaluate using a calculator

θ = 57o Our Solution

If you have two sides of a triangle, we can easily calculate their ratio and then use one of the inverse trig functions to find the missing angle.

18

Example:

Find the indicated angle:

From the angle 𝛼, the given sides are the opposite (5) and the adjacent (3) The trig function that uses opposite and adjacent is the tangent As we are looking for an angle we will use the inverse tangent.

tan−1 (5

3) Sine is opposite over hypotenuse, use inverse to find angle.

sin−1(1.667) Evaluate fraction, take sine inverse using a calculator. 59o Our solution. Example:

Find the indicated angle:

We have one angle and one side. We can use these to find either other side. We will find the other leg, the adjacent side to the 35◦ angle. The 5 is the opposite side, so we will use the tangent to find the leg.

𝑡𝑎𝑛 (35o) = 5/𝑥 Tangent is opposite over adjacent (0.700)𝑥 = 5 Evaluate tangent, solve for x 𝑥 = 7.1 Our solution. 𝑎2 + 𝑏2 = 𝑐2 We can now use Pythagorean theorem to find hypotenuse, c 52 + 7.12 = 𝑐2 Evaluate exponents

25 + 50.41 = 𝑐2 Add, and evaluate for c by taking the square root of both sides 𝑐 = 8.7 The hypotenuse

19

Inverse Trigonometry Functions Activity Find each angle measure to the nearest degree.

1) sin(Z) = 0.4848

2) sin(Y) = 0.6293

3) sin(Y) = 0.6561 4) cos(Y) = 0.6157

Find the measure of the indicated angle to the nearest degree.

20

Directions: Answer the following problems using their associated diagrams, show all you work and include units when necessary. (Diagrams are not drawn to scale) 1. Using the triangle to the right,

a. Calculate the length of side c in two different ways.

b. Determine the angle(s) 𝜃 and 𝛽.

2. Using the triangle to the right, determine the following;

a. Side x (in cm)

b. Side y (in m)

3. Using the triangle to the right find all unknown values.

4. The diagram to the left illustrates the path a raft would follow

across a 40 meter wide river that has a strong upward current.

(The current moves towards the top of the page)

a. What does the hypotenuse represent?

b. How far down the river does the raft travel?

c. How far is the raft displaced from its starting point?

45o

Application of Trigonometry

c 4 cm

3 cm

𝛽

𝜃

10.0 m

y

x

𝛼

35o

z 33 m

56 m

𝜗

𝜃

21

Vectors: When a physical quantity is described by a simple number (magnitude), we call is a scalar quantity. In contrast, a vector quantity has both a magnitude (the “how much” or “how big” part) and a direction in space. Examples of vector quantities include; velocity, acceleration, and force. The velocity vector shown below has a magnitude of 12 m/s and a direction of 50o. A vector quantity is indicated by a special

accent that is an arrow above its variable as shown here: 𝑟 or 𝐴.

All vectors can be broken up into vertical (generally written as ry) and horizontal (generally written as rx) components. These can be found by making a right triangle with the vector as the hypotenuse. Recall that:

sin 𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 and cos 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

Thus for this example:

Thus, it can now be seen that an object moving 12 m/s at 50o is simultaneously moving 9.2 m/s vertically

and 7.7 m/s horizontally.

You can double check your answer(s) for rx and ry by applying the Pythagorean theorem. 𝑎2 + 𝑏2 = 𝑐2 Where a and bare legs of a triangle and c is its hypotenuse.

(𝑟𝑥)2 + (𝑟𝑦)2

= 𝑟2 By simply relabeling the variables we can see how this relates to vectors

and their components

√(7.7𝑚

𝑠)

2

+ (9.2𝑚

𝑠)

2

= |𝑟| Solving the equation for 𝑟 allows us to determine the absolute value of

the vectors magnitude (this is the absolute value because the result must be a real number.

𝑟𝑦 = 9.2𝑚

𝑠 𝑟𝑥 = 7.7

𝑚

𝑠

|𝑟| = 12𝑚

𝑠

x

y

50o

12 m/s

sin(50o) = 𝑟𝑦

12 𝑚/𝑠

𝑟𝑦 = 12𝑚

𝑠𝑠𝑖𝑛(50o)

cos(50o) = 𝑟𝑥

12 𝑚/𝑠

𝑟𝑥 = 12𝑚

𝑠 𝑐𝑜𝑠(50o)

x

y

50o

12 m/s

ry

rx

22

𝑟𝑥

𝑟𝑦

x

y

60o

5 m/s

120o

Vector components: By definition each component vector lies along one of the two coordinate axis direction. Given by the equation(s)

𝑟𝑥 = |𝑟|𝑐𝑜𝑠(𝜃) 𝑟𝑦 = |�⃑�|𝑠𝑖𝑛(𝜃)

Thus, we need only a single number to describe each component. When determining a component vector we must always indicate its direction relative to the positive x–axis. By doing this we make our measurements uniform, and we also communicates to your calculator which direction and quadrant each component vector is in (+ or - ). Determining Vector Angles: When calculating angles it is important to realize that calculators cannot determine what quadrant a vector is in. Calculators simply measure an angle from the closest x axis. Therefore you must determine the quadrant a vector lies in a correct the angle measurement.

(as shown to the left)

Calculations: 1. Find the horizontal and vertical components of the velocity vector shown below; also calculate

the magnitude and direction from the component vectors you calculate.

𝑟𝑦 = __________ 𝑟𝑥 = __________

𝑟 ⃗⃗⃗= __________ 𝜃 = __________

23

𝑟𝑦 = __________ 𝑟𝑥 = __________

𝑟 ⃗⃗⃗= __________ 𝜃 = __________

Notations: There are two types of notations used when identifying vectors. Each notation provides different information about the vector.

Standard notation: This type of notation provides you with the magnitude of the vector as well as the direction in which it is facing. This notation usually takes the form of

o (Magnitude) at (angle) o 53 m at 180 o

Unit vector notation: provides you with the vector component of a respective vector and applies the use of what are called “unit vectors.” A unit vector is a unitless vector that point entirely along an axis (such as the x or y). These unit vectors are �̂� 𝑎𝑛𝑑 �̂�.

o 𝑟 = 𝑟𝑥 �̂� ± 𝑟𝑦 �̂�

o A = 12 m/s 𝑥 + 3 m/s �̂�

It is possible to write express a vector written in standard notation, in unit vector notation. This is simply done by determining the component vectors using trig functions.

Example: Express the vector below in unit vector notation, 53 m at 180o

𝑟𝑥 = |53 𝑚|𝑐𝑜𝑠(180o) = −53 𝑚 Determine the x component of the vector (𝑟𝑥) (As seen on page 22)

𝑟𝑦 = |53 𝑚|𝑠𝑖𝑛(180o) = 0 𝑚 Determine the y component of the vector (𝑟𝑦) (As seen on page 22)

𝑟 = −53 𝑥 + 0 �̂� Rewrite the vector using the notation above (including the unit

vectors)

√(−53𝑚)2 + (0 𝑚)2 = 53 𝑚 To obtain the magnitude of this vector you must apply the Pythagorean theorem. (as seen on page 21)

24

Vectors Answer the following questions, show all your work, circle your answer and remember your units! 1. Break the Following vectors into their horizontal component (𝑟𝑥) and vertical Component(𝑟𝑦) and

write the vector in unit vector notation.

A. 45 m/s at 225o

B. 67 km at 56o

2. Break the Following vectors into their horizontal (X) component and (Y) vertical Component.

A. 57 N at 216o

B. 123 N at 337o

3. Break the Following vectors into their horizontal (X) component and (Y) vertical Component.

A. 45 m/s at 0o

B. 145 m/s at 45o

C. 23 m/s at 50o

25

Vector Addition

Vectors like other mathematical quantities may be added, subtracted or multiplied. Thus vectors have certain properties that apply to them.

Vector Properties: The following are vectors (Shown in unit vector notation): o �⃗⃗� = 𝑢𝑥 �̂� + 𝑢𝑦 �̂� o �⃗⃗⃗� = 𝑤𝑥 �̂� + 𝑤𝑦 �̂� o �⃗� = 𝑣𝑥 �̂� + 𝑣𝑦 �̂�

Associative property u + (v + w) = (u + v) + w The order in which the operations are performed does not matter

Commutative property u + v = v + u Changing the order of the operands does not change the result

Distributive property a(u + v) = au + av Scalar values (such as a) may be multiplied across a set of parentheses

Addition of vectors

Two vectors can be added together to determine the result (or resultant). When adding vectors you must resolve them into their component vectors (In the x and y direction) and add those components to find the resultant vectors components. You cannot simply add the magnitudes of vectors. There are two ways in which we can add vectors, Graphically and Analytically (Using Trigonometry) These methods are outlined on the next two pages.

Example: Add vectors with their components; 𝑣⃗⃗⃗ ⃗, �⃗⃗� and �⃗⃗⃗�

𝑟 = 𝑣⃗⃗⃗ ⃗ + �⃗⃗� + �⃗⃗⃗� Substitute the vectors with their components.

𝑟 = (𝑣𝑥�̂� + 𝑣𝑦�̂�) + (𝑢𝑥�̂� + 𝑢𝑦�̂�) + (𝑤𝑥�̂� + 𝑤𝑦�̂�) Separate the component vectors into their individual direction (x and y).

𝑟 = (𝑣𝑥 + 𝑢𝑥 + 𝑤𝑥)�̂� + (𝑣𝑦

+ 𝑢𝑦 + 𝑤𝑦)�̂� Add the values of the x and y components which will provide you with the resultant vectors components.

𝑟 = (𝑟𝑥)�̂� + (𝑟𝑦

)�̂� Now we have the component of the resultant vector we can find the magnitude and direction of the resultant.

|𝑟| = √(𝑟𝑥)2 + (𝑟𝑥)2 Determine the magnitude of the resultant.

𝜃 = tan−1 (𝑟𝑦

𝑟𝑥)

Determine the angle (direction) of the vector using the inverse tangent function.

26

1. 𝐴 ⃗⃗⃗⃗ = 2.0 𝑚 𝑎𝑡 𝜃1 = 45o

𝐵 ⃗⃗⃗⃗ = 3.0 𝑚 𝑎𝑡 𝜃2 = 135o

2. a. X components using, 𝑟𝑋 = |𝑟| cos 𝜃

𝐴𝑋 = |𝐴| cos 𝜃

𝐴𝑋 = 2.0 𝑚 cos(45𝑂) = 1.414𝑚

𝐵𝑋 = |𝐴| cos 𝜃

𝐵𝑋 = 3.0 𝑚 cos 135𝑂 = −2.121𝑚

b. Y components using, 𝑟𝑌 = |𝑟| sin 𝜃

𝐴𝑌 = |𝐴| sin 𝜃

𝐴𝑌 = 2.0 𝑚 sin(45𝑂) = 1.41𝑚

𝐵𝑌 = |𝐴| sin 𝜃

𝐵𝑌 = 3.0 𝑚 sin 135𝑂 = 2.121𝑚 3.

a. 𝑟𝑋 = 𝐴𝑋 + 𝐵𝑋

1.414 𝑚 − 2.121 𝑚 = −0.707 𝑚

b. 𝑟𝑌 = 𝐴𝑌 + 𝐴𝑌

1.414 𝑚 + 2.121 𝑚 = 3.54 𝑚

4. Determine the magnitude of the Resultant

Vector

c = √𝑎2 + 𝑏2

c = √𝑟 ⃗⃗⃗𝑋2

+ 𝑟 ⃗⃗⃗𝑌2

3.60 m = √(−0.71 𝑚)2 + (3.53 𝑚)2

5. Direction / Angle.

Second Quadrant:

𝜃 = tan−1 (𝑟𝑌

𝑟𝑋) + 180𝑂

1. Clearly determine the:

a. Magnitude of the Vectors.

b. Direction of the Vectors.

2. Break Each Vector into its:

a. X components using,

𝑟𝑋 = |𝑟| cos 𝜃

b. Y components using,

𝑟𝑌 = |𝑟| sin 𝜃

3. Sum up all of the components in each

direction (remember to apply + and - )

a. Sum all the X components

𝑟𝑋 = 𝑟1,𝑋 + 𝑟2,𝑋 + ⋯

b. Sum all the Y components

𝑟𝑌 = 𝑟1,𝑌 + 𝑟2,𝑌 + ⋯

4. Use the Pythagorean theorem to

determine the magnitude of the Resultant

Vector

c = √𝑎2 + 𝑏2

5. To determine the direction of the

Resultant Vector in reference to the

Positive X axis we must use the

Arc Tangent.

𝜃 = tan−1 (𝑟𝑌

𝑟𝑋)

Adding Vectors Mathematically

27

Always add vectors graphically by orienting them head – tail (the head of a vector is the arrow). The first vector should always start at the origin. Determine a scale factor (if needed) to make the graph easier to construct.

1.

𝐴 ⃗⃗⃗⃗ = 2.0 𝑚 𝑎𝑡 𝜃1 = 45𝑂

𝐵 ⃗⃗⃗⃗ = 3.0 𝑚 𝑎𝑡 𝜃2 = 135𝑂

2.

3.

4. This problem only has two vectors.

5.

1. Clearly determine the:

a. Magnitude of the Vectors.

b. Direction of the Vectors.

2. Draw the first Vector to scale on the

coordinate system.

a. Make sure that the Vector is in the

correct direction and quadrant.

3. Draw a mini coordinate system at the head

of the first Vector.

a. Draw the next vector head to tail of

the last vector.

4. Repeat step 3 for each Vector…

5. After drawing all vectors, draw the

Resultant Vector from the origin to the head

of the last vector.

6. To determine the magnitude of the resultant

measure its length and apply the scale you

chose.

7. To determine the angle / direction use a

protractor to measure it.

Adding Vectors Graphically

𝜃1

𝐴

𝜃1

𝜃2

𝐴

�⃗⃗�

𝐶

𝜃1

𝜃2

𝐴

�⃗⃗�

28

Vector Addition 1. Add the following vectors below analytically, and then graphically:

A. 𝐴 = 5 𝑚

𝑠 �̂� + 0

𝑚

𝑠 �̂�

B. �⃗⃗� = 0 𝑚

𝑠 �̂� + 4.95

𝑚

𝑠 �̂�

2. Add the following vectors below analytically, and then graphically:

A. 𝐶 = 8.50 𝑚 𝑎𝑡 90o

B. 𝐷 = 6.0 𝑚 𝑎𝑡 180o

3. Add the following vectors below analytically, and then graphically:

A. �⃗⃗� = −4.50 𝑘𝑚 �̂� − 5.63 𝑘𝑚 �̂�

B. �⃗� = 9.0 𝑘𝑚 𝑎𝑡 0o

𝑟𝑥 = __________

𝑟𝑦 = __________

|𝑟|= __________

𝜃 = __________

𝑟𝑥 = __________

𝑟𝑦 = __________

|𝑟|= __________

𝜃 = __________

𝑟𝑥 = __________

𝑟𝑦 = __________

|𝑟|= __________

𝜃 = __________