St Patrick’s College
YEAR 11
INTRODUCTORY
PHYSICS
SCIENTIFIC THINKING
MEAUSREMENT
DIMENSIONAL ANALYSIS
UNCERTAINTY AND ERROR
GRAPHICAL ANALYSIS
ALGEBRA AND TRIGONOMETRY
VECTORS
BASIC KINEMATICS
RECTILLINEAR MOTION
NEWTON’S LAWS
Page 1
Science
Science (from the Latin scientia, ‗knowledge‘), in the broadest sense, refers to
any systematic knowledge or practice. In a more restricted sense, science refers
to a system of acquiring knowledge based on the scientific method, as well as to
the organized body of knowledge gained through such research.
The scientific method seeks to explain the events of nature in a reproducible
way, and to use these reproductions to make useful predictions. It is done through observation of natural
phenomena, and/or through experimentation that tries to simulate natural events under controlled conditions.
It provides an objective process to find solutions to problems in a number of scientific and technological
fields. A strict following of the scientific method attempts to minimize the influence of a scientist's bias on
the outcome of an experiment. This can be achieved by correct experimental design, and a thorough peer
review of the experimental results as well as conclusions of a study.
Scientists use models to refer to a description or depiction of something, specifically one which can be used
to make predictions that can be tested by experiment or observation. A hypothesis is a contention that has
been neither well supported nor yet ruled out by experiment. A theory, in the context of science, is a
logically self-consistent model or framework for describing the behavior of certain natural phenomena. A
theory typically describes the behavior of much broader sets of phenomena than a hypothesis—commonly, a
large number of hypotheses may be logically bound together by a single theory. A physical law or law of
nature is a scientific generalization based on a sufficiently large number of empirical observations that it is
taken as fully verified.
Isaac Newton's law of gravitation is a famous example of an established law that was later found not to be
universal—it does not hold in experiments involving motion at speeds close to the speed of light or in close
proximity of strong gravitational fields. Outside these conditions, Newton's Laws remain an excellent model
of motion and gravity. Since general relativity accounts for all the same phenomena that Newton's Laws do
and more, general relativity is now regarded as a more comprehensive theory.
Mathematics is essential to many sciences. One important
function of mathematics in science is the role it plays in the
expression of scientific models. Observing and collecting
measurements, as well as hypothesizing and predicting, often
require mathematical models and extensive use of analysis.
Mathematical branches most often used in science include
calculus and statistics, although virtually every branch of
mathematics has applications, even "pure" areas such as
number theory and topology. Mathematics is fundamental to
the understanding of the natural sciences and the social
sciences, all of which rely heavily on statistics. Statistical Velocity-distribution data of a gas of
methods, comprised of accepted mathematical formulas for rubidium atoms, confirming the discovery of
summarizing data, allow scientists to assess the level of a new phase of matter, the Bose–Einstein
reliability and the range of variation in experimental results. condensate.
Physics (from the Greek physis, ‗nature‘), is the scientific study of the properties of matter and energy.
‗Matter‘ refers to any materials which occupy space and can be examined by measuring or experimental
testing. ‗Energy‘ is something a body possesses which enables it to do ‗work‘ and/or change. More
technically, it is the general analysis of nature that aims to understand how the world around us behaves.
Advances in physics often translate to the technological sector, and sometimes influence the other sciences,
as well as mathematics and philosophy. For example, advances in the understanding of electromagnetism
have led to the widespread use of electrically driven devices (televisions, computers, home appliances etc.);
advances in thermodynamics led to the development of motorized transport; and advances in mechanics led
to the development of the calculus, quantum chemistry, and the use of instruments like the electron
microscope in microbiology.
Page 2
Although physics encompasses a wide variety of phenomena, all physicists are
expected to be familiar with the basic theories of classical mechanics,
electromagnetism, relativity, thermodynamics, and quantum mechanics. Each of
these theories has been tested in numerous experiments and proven to be an accurate
model of nature within its domain of validity.
Some of the greatest physicists of all time were: Sir Isaac Newton (1643-1727),
Albert Einstein (1879-1955), Archimedes (287BC-212BC), James Clerk Maxwell
(1831-1879), Niels Bohr (1885-1962), Werner Heisenberg (1901-1976), Galileo
Galilei (1564-1642), Richard Feynman (1918-1988), Paul Dirac (1902-1984), Erwin
Schrödinger (1887-1961) and Ernest Rutherford (1871-1937).
Physics and Chemistry are usually prerequisite fields of study for electrical, mechanical, materials, industrial
and aeronautical engineers as well as for chemists, environmental scientists, physicists, computer scientists,
metallurgists, medical scientists, meteorologists, geologists, agricultural scientists, dentists, physiotherapists,
optometrists, pharmacists, pilots, astronauts and veterinarians.
Questions:
1. What is ‗science‘?
2. What is ‗physics‘?
3. From which languages do the words ‗science‘ and ‗physics‘ originate?
4. What is the scientific method?
5. How does a hypothesis differ from a theory?
6. How does a theory differ from a law?
7. Why is mathematics so important in studying scientific fields?
8. What are the primary fields of physics?
9. Give two examples of how advances in physics have led to better technology.
Research topics:
1. Investigate the contributions made by one of the above physicists.
2. Investigate one of the scientific career options listed above.
Page 3
Scientific Thinking 1
Arguably, one of the most significant dates of the of the 20 th Century was July 20th, 1969. On this day, Neil
Armstrong became the first person to walk on the Moon. However, there are people who believe otherwise.
There are claims that the moon landings were faked. There have been many books and television programs
made about this conspiracy. Why? There are several psychological, social and cultural reasons why
conspiracy theories are popular. However, when it comes to the fake moon landing conspiracy, the claims
are generally easily dismissed with basic science.
Explain why each of the following claims is scientifically faulty.
1. In the photos taken on the moon there are no stars.
With no atmosphere, the sky would be black and the
stars should be even brighter than they appear on Earth.
In the picture below there are no stars – it must be fake.
2. There is only one light source on the moon – the sun.
In area B in the photo, the astronaut is facing away
from the sun, so it should be in complete shadow. Yet
we can see him clearly. Obviously this is a result of
extra lighting on the set.
3. Looking at area C you will notice that the surface of the
moon fades off into the distance, then is met with the
moon's horizon. In a no-atmosphere environment, the
ground shouldn't have faded out, but stayed crystal
sharp unto the moon's horizon.
4. In any case, average day temperatures on the moon are
about 280ºF. At this temperature the film would crinkle
up and be useless. Clearly the photo was not taken on
the moon.
5. 100 miles above the Earth there is a radiation belt call
the Van Allen belt. This intense radiation would have
killed the Astronauts if they travelled through it.
6. There are millions of small meteors in space that travel
at millions of miles per hour. They would have
destroyed the ship on the way to the moon.
7. How are the footprints made? In order for footprints or tyre tracks to be left, the sand or dust would
need to have water in it or it would not make so clearly defined lines.
8. The photo of the American flag shows that it is
fluttering. But how can this be if there is no air on
the moon?
9. In the picture on the right, the shadows are not
parallel. As there is only one light source on the
moon, this wouldn‘t happen. Therefore it must
have been taken in a studio with different light
sources.
Page 4
Scientific Thinking 2
You are a manufacturer of boat sails. A famous sailor wants to build a new yacht for the upcoming Sydney
to Hobart yacht race. He asks you to make a sail for his boat from a choice of eight different fibres. The
following tables show the different properties of the available fibres.
Quantitative properties of available fibres:
A B C D E F G H
Density
(g/cm³)
1.31 1.55 1.34 1.52 1.30 1.14 1.38 1.16
Dry strength
(N/m× 104)
1.1 3.2 3.6 2.4 1.2 4.5 4.1 2.4
Wet strength
(N/m × 104)
0.9 3.4 2.6 1.3 0.8 3.8 4.1 2.0
% stretch
before 30-45 5-10 20-25 17-25 25-28 23-42 25-35 25-45
breaking
Force needed
to stretch 29 53 63 64 36 20 60 40
(N/m × 104)
% elastic
recovery
100 70 90 60 85 100 80 90
Moisture
absorbance 16 9 11 13 3 5 0.5 2
(%)
Qualitative properties of available fibres:
A warm feel, may shrink a lot, moderate wear, can irritate skin
B firm, comfortable, tough, may shrink a little, may crease
C crisp attractive feel, very poor resistance to wear
D soft, very good resistance to wear, crease-resistant
E crease-resistant
F soft, hard-wearing, crease-resistant, very tough
G hard-wearing, very tough
H soft, warm feel, non-shrink, tough, crease-resistant
Questions:
1. Which of the available fibres has the better qualitative properties?
2. For each of the quantitative properties, determine the ideal value.
3. Which of the available fibres has the better quantitative properties?
4. Which of the available fibres would you recommend to the sailor?
Page 5
Measurement
Making observations is a fundamental skill of life. A quantitative observation, or measurement, always
consists of two parts: a number and a unit. Both parts together make the measurement meaningful. Scientists
recognised long ago that a standard system of units had to be adopted if measurements were to be useful in a
society. However, different systems of units were developed in different parts of the world. The two major
systems are the Imperial system and the metric system.
In 1960, an international agreement set up a system of units called the International System (SI). This
system is based on the metric system. There are seven fundamental quantities from which all other quantities
are derived:
Physical Quantity Name of Unit Unit symbol
Mass kilogram kg
Length metre m
Time second s
Temperature Kelvin K
Electric current ampere A
Amount of a substance mole mol
Luminous intensity candela cd
The metre is defined as the distance light travels, in a vacuum, in one 299792458 th of a second. The
kilogram is the mass of a platinum-iridium cylinder kept at Sevres in France. It is the only basic unit still
defined in terms of a material object, and also the only one with a prefix already in place.
The second is defined as the length of time taken for 9192631770 periods of vibration of the caesium-133
atom to occur. The ampere is the current that produces a specified force between two parallel wires which
are 1 metre apart in a vacuum. The Kelvin is one 273.16 th of the thermodynamic temperature of the triple
point of water.
The mole is the amount of substance that contains as many elementary units as there are atoms in 0.012 kg
of carbon-12. The candela is the intensity of a source of light of specified frequency, which gives a specified
amount of power in a given direction.
Some of the more common derived units are:
Physical Quantity Name of Unit Unit symbol
Frequency hertz Hz
Energy (work) joule J
Force newton N
Electric charge coulomb C
Pressure pascal Pa
Power watt W
2 2
Area metre m
3 3
Volume metre m
-1
Velocity metre per second ms
2 -2
Acceleration metre per second ms
3 -3
Density kilogram per metre kg m
Capacity litre L
Capacitance farad F
Potential difference volt V
Page 6
Because using the fundamental units are not always convenient, prefixes are used to change the size of the
unit. The following are the prefixes used in the metric system.
Prefix Symbol Value Meaning
24
yotta Y 10 septillion
21
zetta Z 10 sextillion
18
exa E 10 quintillion
15
peta P 10 quadrillion
12
tera T 10 trillion
9
giga G 10 billion
6
mega M 10 million
3
kilo k 10 thousand
2
hector h 10 hundred
1
deca da 10 ten
-1
deci d 10 tenth
-2
centi c 10 hundredth
-3
milli m 10 thousandth
-6
micro µ 10 millionth
-9
nano n 10 billionth
-12
pico p 10 trillionth
-15
femto f 10 quadrillionth
-18
atto a 10 quintillionth
-21
zepto z 10 sextillionth
-24
yocto y 10 septillionth
There are several conventions that need to be known for using SI units and prefixes:
All symbols for units should be written in an upright Roman font – so as to distinguish from
mathematical variables which are written in italics (eg: m for metres, m for mass).
All symbols are written in lowercase unless derived from the name of a person (eg: the unit for pressure is
named after Blaise Pascal, so its symbol is written Pa, but the unit is written ―pascal‖).
Any unit may only have one prefix (eg: you cannot write millimillimetre).
Units are not pluralized (eg: 25 kg , not 25 kgs).
A space always separates the number and the symbol (eg: 25 kg , not 25kg). The exceptions are the
imperial units for angles: degrees, minutes and seconds (° ‗ ―).
A space or centre-dot is used to join symbols for derived units formed using multiplication (eg: N m, or
N·m, but not Nm)
A solidus can be used to join symbols for derived units formed by division (eg: m/s). However, the
preferred method is to use index notation as it avoids ambiguity for complex derived units (eg: m·s -1,
kg·m-1·s-2
It is also important to note the SI preferred way of showing a decimal fraction is to use a common on the line
to separate the whole number from its fractional part. However, it is accepted in English-speaking countries
to use a point on the line instead (eg: twelve and one half can be written 12,5 or 12.5 but not 12·5)
The SI method for writing large numbers allows using spaces between digits but not commas or points.
(eg: seven billion is written: 7000000000 or 7 000 000 000 but not 7,000,000,000)
Page 7
Dimensional Analysis
To convert a given result from one unit to another is done by the unit factor method:
Use the equivalence statement that relates the two units.
Derive the appropriate unit factor by looking at the direction of the required change.
Multiply the quantity by the unit factor to cancel the unwanted units.
Some useful conversion factors:
2.54 cm = 1 in inch
-10
1×10 m =1 Å angstrom
8
1.4959787×10 km = 1 AU astronomical units
1.8288 m = 1 fm fathom
30.48 cm = 1 ft foot
15
9.460730473×10 m = 1 ly light year
1.609344 km = 1 mi mile
1852 m = 1 NM nautical mile
91.44 cm = 1 yd yard
2
4046.86 m = 1 ac acre
2
10000 m = 1 ha hectare
3
1 dm = 1 L litre
5 mL = 1 tsp teaspoon
15 mL = 1 tbsp tablespoon
250 mL = 1 cup cup
3.785 L = 1 gal (US) US gallon
4.546 L = 1 gal (Imp) Imperial gallon
200 mg = 1 kt carat
28.35 g = 1 oz ounce
0.4536 kg = 1 lb pound
6.35 kg = 1 st stone
1000 kg = 1 t tonne
907.185 kg = 1 tn US ton
1016.047 kg = 1 tn Imperial ton
365 d = 1 y calendar year
365.2563 d = 1 y sidereal year
4.184 J = 1 cal calorie
4.184 kJ = 1 Cal Calorie
4.184 GJ = 1 tTNT ton of TNT
Eg: To convert 35 mi/gal into km/L:
35 1.609344 1
15
1 1 3.785
Therefore 35 mi/gal = 15 km/L.
Page 8
Uncertainty in measurement
It is very important to realise that any measurement always has some degree of uncertainty. The uncertainty
depends on the precision of the measuring device.
For example, using a bathroom scale you might find that two objects both measure 1.5 kg and conclude that
they have the same mass. However, when you use a highly precise balance scale you find that their masses
are 1.476 kg and 1.518 kg. What do you conclude?
In the first case, the uncertainty occurs in the tenths place; in the second case, the uncertainty occurs in the
thousandths place. Do the objects have the same mass? The answer depends on how accurate you want to be
and the certainty of the measurements. For this reason, it is important to indicate the uncertainty in
measurements. These numbers are called the significant figures of a measurement. The uncertainty in the
last digit is usually assumed to be ±0.5 unless otherwise indicated. For example, 1.86 kg is generally taken
to mean 1.86 ± 0.005 kg.
Eg: Two students are asked to measure an amount of liquid. The first uses a pipet and measures 25.00 mL.
The second uses a graduated cylinder and measures 25 mL. Did they get the same result? No, the quantity
25 mL means the volume is between 24.5 mL and 25.5 mL. Whereas the quantity 25.00 mL means that the
volume is between 24.995 mL and 25.005 mL.
Rules regarding Significant Figures
Nonzero digits always count as significant figures. (Eg: 256 has three significant figures)
Leading zeros do NOT count as significant figures. (Eg: 0.025 has two significant figures)
Trailing zeros do NOT count as significant figures unless the number contains a decimal point.
Captive zeros do count as significant figures. (Eg: 1.008 has four significant figures)
Exact numbers can be assumed to have an infinite number of significant figures.
(Eg: The 2 and the π in 2πr and the 2.54 in the definition 2.54 cm = 1 in are all exact numbers)
For addition and subtraction, the result has the same number of decimal places as the least precise
measurement used in the calculation.
For multiplication or division, the result has the same number of significant figures as the least precise
measurement used in the calculation.
In a series of calculations, carry the extra digits through to the final result THEN round appropriately.
Questions:
1. What are SI units?
2. What are the seven fundament quantities?
3. What are the seven fundament units?
4. What are the units for
a. Force
b. Power
c. Energy
d. Pressure
5. How many times larger is a megametre than a micrometre?
6. Explain the difference between the symbols ‗m‘ and ‗m‘.
7. Which is longer, a mile or a nautical mile?
8. What is the percentage difference between a calendar year and a sidereal year?
9. The circumference of a circle that has a measured radius of 2.5 centimetres, is
found by multiplying 2, π and 2.5. Which of these three numbers has the least
precision?
Page 9
Exercises:
1. Which of the following are exact numbers?
a. Brisbane has a population of 1.2 million people.
b. There are 12 eggs in a dozen.
c. One yard is equal to 0.9144 metres.
d. The attendance at a football game was 52 806.
e. The budget deficit of the United States was $559 billion last year.
2. How many significant figures are in each of the following?
a. 12 e. 1.01×10-6
b. 1098 f. 1000.
c. 2.001×103 g. 22.04030
d. 0.00000101
3. Round each of the following numbers to three significant figures.
a. 312.54 d. 0.31254
b. 0.00031254 e. 31.254×10-3
c. 31254000
4. Perform the following mathematical calculations and express the result to the
correct number of significant figures.
a. 97.381 4.2502 0.99195
b. 4.184 100.62
c. 9.04 8.23 21 .954 81 .0 3.1416
d. 6.022 10 23 1.05 10 2
5. Perform each of the following conversions.
a. 8.43 cm to millimetres d. 908 oz to kilograms
2
b. 2.41×10 cm to metres e. 12.8 L to gallons
c. 294.5 nm to centimetres f. 4.48 lb to grams
6. A marathon race is 26 miles, 385 yards. What is this distance in kilometres?
7. Perform each of the following conversions.
a. 125 mi/h to km/h c. 2.70 g/cm3 to kg/m3
b. 225 km/h to mi/h d. 2.70 g/cm3 to lb/ft3
3
8. If the density of diamond is 3.51 g/cm , what is the volume of a 5.0 carat diamond?
9. The density of pure silver is 10.5 g/cm3 at 20°C. If 5.25 g of silver pellets is added
to a graduated cylinder containing 11.2 mL of water, to what volume level will the
water in the cylinder rise?
10. A cylindrical bar of gold that is 1.5 in high and 0.25 in in diameter has a mass of
23.1984 g, as determined on an analytical balance. An empty graduated cylinder is
weighed on a triple beam balance and has a mass of 73.47 g. After pouring a small
amount of liquid into the graduated cylinder, the mass is 79.16 g. When the gold
cylinder is placed in the graduated cylinder the liquid covers the top of the gold
cylinder and the volume indicated on the graduated cylinder is 8.5 mL. What is the
density of gold?
Page 10
Graphing
Representing data or relationships between variables graphically is very useful in all fields of study. It is
important to be able to recognize and understand some of the more common types of mathematical
relationships:
Directly Proportional Linear
25 30
25
20
20
15
15
y
y
10
10
5 5
0 0
0 2 4 6 8 10 0 2 4 6 8 10
x x
Inverse Quadratic
16 70
14 60
12
50
10
40
8
y
y
30
6
4 20
2 10
0 0
0 2 4 6 8 10 0 2 4 6 8 10
x x
Exponential Periodic
2000 3.5
1800
3
1600
1400 2.5
1200 2
1000
y
y
800 1.5
600 1
400
0.5
200
0 0
0 2 4 6 8 10 0 2 4 6 8 10
x x
Page 11
Questions:
1. Plot the following data points:
x 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
y 1.2 1.6 2.1 2.5 2.8 3.3 3.7 4.1
What type of relationship exists?
Predict the value of y when x 0.9
2. Plot a graph of y vs x
x 0.0 2.5 3.6 5.4 7.7 8.3 9.5 11.1
y 0.0 3.1 6.5 14.6 29.6 34.5 45.1 61.6
What type of relationship appears to exist?
Plot a graph of √y vs x to confirm
3. Plot of a graph of y vs x
x 0.1 0.2 0.4 0.5 0.7 0.8 0.9 1.1
y 31.5 16.1 8.0 6.4 4.6 4.0 3.5 2.9
What type of relationship appears to exist?
1
Plot a graph of y vs to confirm
x
4. Plot of a graph of y vs x
x 0.1 0.9 1.6 2.4 3.2 3.9 4.7 5.1
y 1.1 1.6 2.7 4.4 6.2 7.6 8.7 8.9
What type of relationship appears to exist?
Can you confirm it by plotting a second graph?
Page 12
Interpreting Graphs
Understanding different types of graphs is another important skill. There are different types of graphs for
different types of data. What types of data are there? Look at each of the following graphs carefully before
answering the questions that accompany it.
Questions:
1. The number of cars speeding at various locations were recorded on a particular
day.The results are shown in the graph below.
a. At which location was the percentage of speeding cars greatest?
b. What was the percentage of speeding cars there?
c. Can we tell how many cars were speeding there?
d. Why has the percentage of speeding cars, rather than number of speeding
cars, been graphed?
e. Why might the percentage of speeding cars be so much larger in locations
1 and 2?
f. Do we know from the graph what the speed limits were in each location?
g. Do we know from the graph by how much the cars exceeded the speed
limit?
2. The chart below displays the climate of Melbourne, based on average monthly
maximum temperatures and average daily rainfall for each month.
Page 13
a. In which month(s) is the average temperature highest and lowest
respectively?
b. In which month(s) is the rainfall highest and lowest respectively?
c. With each of the seasons, we expect particular characteristics. Eg: in
summer, we expect high temperatures and low rainfall. What are the
characteristics of:
i. autumn (March, April, May)?
ii. winter (June, July, August)?
iii. spring (September, October, November)?
3. In their physical education class the girls were asked to sprint for 10 seconds. The
teacher recorded their results on 2 different days. They are shown below.
a. Why are there 2 columns for each girl?
b. Which girl ran the fastest on either day?
c. How far did she run on each day?
d. Which girl improved the most?
e. Were there any students who did not improve? Who were they?
f. Could this graph be misleading in any way? Explain your answer.
g. Why might the graph‘s vertical axis start at 30 m?
4. The diagram below shows the heights of a group of students.
a. What is the interquartile range?
b. Which of the following is not true?
i. 50% of the students are shorter than 1.56 m.
ii. The number of students shorter than 1.48 m is less than the number
of those taller than 1.60 m.
iii. The range of the heights of the students in the group is 34 cm.
iv. 75% of students have a height of 1.6 m or under.
Page 14
Algebra Review
Mathematics provides the basic tools for developing models and analysing results in all fields of study. In
order to be able learn concepts in physics and chemistry, basic algebraic skills are required.
Questions:
1. Rearrange the following formulae to make the symbol in brackets the subject.
s f. W Fs (F)
a. v (t)
t g. KE 1 mv 2 (v)
2
b. F ma (m) m
c. P mv (v) h. T 2 (k)
k
m
d. D (m) v2
v i. a (v)
v r
e. a (v) j. Q mcT (c)
t
2. Rearrange the following formulae to make the symbol in brackets the subject.
a. Mass energy equivalence: E mc 2 (c)
L
b. Brightness of a star: b (d)
4d 2
v
c. Doppler shift: (λ)
c
d. Luminosity: L 4R 2T 4 (T)
1 1 1
e. Synodic period: (E)
P E S
3kT
f. Velocity of a particle: v (m)
m
GM
g. Velocity of a satellite: v (r)
r
Gm1 m2
h. Universal gravitation: F (d)
d2
d Lb0
i. Distance to a star: (L)
d0 L0 b
2GM E md
j. Tidal forces: Ft (r)
r3
4 2 a 3
k. Orbit equation: G (m m ) ( m1 )
P2
1 2
T0
l. Relativity: T (v)
1 v
2
c
Page 15
Trigonometry Review
Mathematics provides the basic tools for developing models and analysing results in all fields of study. In
order to be able learn concepts in physics and chemistry, basic trigonometric skills are required.
Questions:
1. Find the indicated side in each of the following:
2. Find the indicated angle in each of the following:
3. Find the value of h and x in the diagram below.
4. Find the two diagonal lengths marked in the diagram above.
5. A hiker travels 20km on a bearing of 120ºT.
a. How far north of her starting point is she?
b. How far east of her starting point is she?
Page 16
Vectors
A vector is a quantity characterized by a magnitude and a direction.
Graphically vectors are represented by arrows. Vectors can be written in
several ways, however, the most commonly accepted forms are to use the
points with an arrow above them, or by using a tilde. For example, the
vector on the right could be named OA or a . Vectors are usually written
~
in terms of their Cartesian values x, y as per the example on the right.
However, they are sometimes written in terms of their magnitude and
direction r , , also called modulus and argument, or in terms of their
unit vectors x i y j . The reason for the different notations is that some
~ ~
forms are easier to deal with for particular types of calculations.
Addition of vectors: Subtraction of vectors: Multiplication:
To numerically operate on vectors, they must be first reduced into their horizontal (i) and vertical (j)
components. These can be added or subtracted easily and then the resultant vector can be found.
Questions:
1. Determine the magnitude and argument of the vector a in the top example.
~
2. Determine the magnitude and argument of the following vectors:
a. a (3,4) e. e a b
~ ~ ~ ~
b. b (5,12) f. f a b
~ ~ ~ ~
c. c 6 i 6 j g. g 2 c 3 d
~ ~ ~ ~ ~
~
d. d 2 i j h. h c 2d
~ ~ ~ ~ ~ ~
3. Add the following vectors together.
a. b.
Page 17
Basic Kinematics
Kinematics is a branch of mechanics which describes the motion of objects without the consideration of the
masses or forces that bring about the motion. In contrast, dynamics is concerned with the forces and
interactions that produce or affect the motion.
Kinematics studies how the position of an object changes with time. Position is measured with respect to a
set of coordinates. Velocity is the rate of change of position. Acceleration is the rate of change of velocity.
Velocity and Acceleration are the two principal quantities which describe how position changes. The
simplest application of kinematics is to point particle motion (translational kinematics or linear kinematics).
The description of rotation (rotational kinematics or angular kinematics) is more complicated.
Questions:
1. Determine the average speeds for the following journeys. (Either km/h or m/s)
a. A truck travels 1800km in 22 hours.
b. John rides his bike for 90min and covers 18km.
c. Paul runs the 400m in 55 sec.
d. A snail takes 25min to travel 3m.
2. Determine the time taken to travel the following distances at the given speeds.
a. Driving to the city (20km) at 55km/h.
b. Driving the 105km to the Gold Coast and back at 95km/h
c. Sprinting 15m at 9m/s.
d. Flying to Sydney (995km) at the speed of sound (340m/s)
3. Determine the distances travelled at the following speeds.
a. Matt drives for 2 hours at 55km/h.
b. Sally runs for 20min at 10km/h
c. Sam swims for 30min at 5km/h
d. An F-111 flys at Mach 2 (twice the speed of sound) for 20min.
4. Penny and Suzy are having a race over 1500m. Penny is the faster runner (4m/s),
so she gives Suzy (who can run at 3.5m/s) a 30sec head-start.
a. Who wins the race?
b. As the winner crosses the finish line, how far behind is the other runner?
5. Laurence averages 100km/h on his trip to the Gold Coast (105km). On his return
trip he averages only 90km/h because of traffic. What is his average speed for the
whole trip?
6. Natasha is an excellent athlete and runs to the local shops, which are 2km away,
and back every morning. On a calm day it takes her 20min. How long would it
take her on a very windy day, if running against the wind her average speed is
reduced by 2km/h and running with the wind increases her average speed by
2km/h?
7. An aeroplane flys at 800km/h for one third of its flught time and averages
700km/h for the entire trip. What is the average speed, in km/h over the remaining
part of the journey?
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8. Jack climbed at a constant speed up the beanstalk. At 2 o‘clock he was one-sixth
of the way up and at 4 o‘clock he was three-quarters of the way up.
a. What fraction of the way up the beanstalk had he climbed at 3 o‘clock?
b. When did he start climbing?
c. When did he reach the top?
9. In a race of 2000m, Raelene finishes 200m ahead of Marjorie, and 290m ahead of
Betty. If Marjorie and Betty continue to run at their previous speeds, by how
many metres will Marjorie finish ahead of Betty?
10. Examine the distance vs time graphs below. They show the speeds of two runners
over 100m.
A B
120 120
100 100
80 80
60 60
40 40
20 20
0 0
0 2 4 6 8 10 12 0 2 4 6 8 10 12
a. Determine the average speeds of both runners?
b. Which runner reached the fastest speed? What was it?
c. Which runner reached the 50m mark first?
d. Which runner won the race?
11. Four runners ran a race over 100m. Use the graphs to determine who won the
race.
A and B C and D
12 14
10 12
10
8
Speed
8
Speed
6
6
4 4
2 2
0
0
0 2 4 6 8 10 12
0 2 4 6 8 10 12
Tim e
Time
A B C D
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Rectilinear Motion
Consider an object which is fired directly upwards and falls back to the ground so that its trajectory is
contained in a straight line. If we adopt the convention that the upward direction is the positive direction,
the object experiences a constant acceleration of approximately -9.81 m/s2. Therefore, its motion can be
modeled with the equations governing uniformly accelerated motion. There are several interesting
kinematic questions we can ask about the particles motion: How long will it be airborne? What altitude will
it reach before it begins to fall? What will its final velocity be when it reaches the ground? For the sake of
example, assume the object has an initial velocity of +50 m/s.
Equations of motion in one-dimension with constant acceleration
An object moving with constant acceleration is said to be undergoing uniformly accelerated motion
(UAM). Its motion can be described with four simple algebraic equations:
v u at v 2 u 2 2as
s ut 1 at 2 s
u v t
2
2
In the formulae u is the objects initial velocity, v is the objects final velocity, t is the time taken, s is the
displacement and a is the acceleration.
Questions:
1. A car accelerates uniformly from rest at 5m/s2 for 10 seconds. What is its final
velocity?
2. What is the acceleration of a vehicle that accelerates uniformly from 10m/s to
20m/s in 4 seconds?
3. A car accelerates uniformly from rest at 2.5m/s2 over a distance of 100m. What is
its final velocity?
4. What is the acceleration of a vehicle that accelerates uniformly from 10m/s to
20m/s over a distance of 10m?
5. A car accelerates uniformly from 2m/s at 5m/s2 for 15s. How far does it travel
during this time?
6. A body falls 15m in 1s. What was its initial velocity?
7. How long does a particle take to accelerate uniformly from 2m/s to 20m/s at
5m/s2?
8. A plane accelerates uniformly for 5 seconds at 5m/s2. If its final velocity is
100m/s, what was its initial velocity?
9. How far does a particle travel when accelerating from 5m/s to 15m/s at 3m/s2?
10. A plane accelerates uniformly over 100m at 8m/s2. If its final velocity is 100m/s,
what was its initial velocity?
11. A body is thrown down a 100m cliff with an initial velocity of 5m/s. How long
does it take to hit the ground?
12. A particle travels 150m in 6s. If its initial velocity was 2m/s, what was its
acceleration?
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Newton’s Laws of Motion
Newton's laws of motion are three physical laws which provide relationships between the forces acting on a
body and the motion of the body, first compiled by Sir Isaac Newton. Newton's laws were first published
together in his work Philosophiae Naturalis Principia Mathematica (1687). The laws form the basis for
classical mechanics. Newton used them to explain many results concerning the motion of physical objects.
In the third volume of the text, he showed that the laws of motion, combined with his law of universal
gravitation, explained Kepler's laws of planetary motion.
Newton‘s first law - The Law of Inertia:
“Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar
as it is compelled to change its state by force impressed.”
The net force on an object is the vector sum of all the forces acting on the object. Newton's first law says
that if this sum is zero, the state of motion of the object does not change. Essentially, it makes the following
two points:
An object that is not moving will not move until a net force acts upon it.
An object that is in motion will not change its velocity until a net force acts upon it.
Newton‘s second law – The Law of Momentum:
“The rate of change of momentum of a body is proportional to the resultant force acting on the body and is
in the same direction.”
Observed from an inertial reference frame, the net force on a particle is proportional to the time rate of
change of its linear momentum: F = d[mv] / dt. Momentum is the product of mass and velocity.
This law is often stated as F = ma (force is equal to its mass multiplied by its acceleration)
Newton‘s third law – The Law of Reciprocal Actions:
“All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.”
Whenever a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the
same magnitude in the opposite direction. The strong form of the law further postulates that these two
forces act along the same line.
To every action force there is an equal, but opposite, reaction force
Questions:
1. Use Newton‘s first law of motion to explain the following:
a. A moving car slows down when left to roll in neutral along a flat road.
b. A bicycle needs to be pedaled even when travelling at a constant speed.
2. What forces bring a bicycle to a stop during braking?
3. Use inertia to explain each of the following:
a. You fall forwards if, while walking, you trip over a log on the ground.
b. A heavy, rolling pram is more difficult to stop that a light rolling pram.
4. Why do you think modern cars have airbags?
5. Find the force required to push a 6 kg skateboard through a distance of 2 m from
rest in 5 s, assuming there is no friction.
6. A train of mass 104 kg moving at 15 ms-1 stops in a distance of 400 m. Calculate
the force applied by the train‘s braking system.
7. A book of mass 1.5 kg exerts a downward force on a table.
a. What is the magnitude of this force?
b. What is the magnitude of the force exerted by the table on the book?
8. Explain how the motion of a rocket taking off is an example of an action-reaction
situation.
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REVISION
1. What is science?
2. What is physics?
3. What is the scientific method?
4. How does a hypothesis differ from a theory?
5. How does a theory differ from a law?
6. Why is mathematics so important in studying science?
7. What are the primary fields of physics?
8. Name two famous physicists.
9. What are SI units?
10. What are the seven fundamental quantities and their respective units?
11. Explain the difference between the symbols ‗m‘ and ‗m‘.
12. Which of the following are exact numbers?
a. Sydney has a population of 3.7 million people.
b. There are 13 eggs in a baker‘s dozen.
c. One yard is equal to 0.9144 metres.
13. Round each of the following numbers to three significant figures.
a. 302.53 c. 0.1254
b. 0.000883 d. 61.54×10-3
14. Perform the following mathematical calculations and express the result to the
correct number of significant figures.
a. 72.31 2.202 0.99357
b. 3.182 106.60
c. 9.04 8.23 21.954 81.0 3.1416
d. 6.022 10 1.53 10
23 2
15. Perform each of the following conversions.
a. 8.43 mm to centimetres c. 294.5 cm to nanometres
b. 2.41×102m to centimetres d. 908 kg to ounces
16. A marathon race is 26 miles, 385 yards. What is this distance in metres?
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17. Perform each of the following conversions.
a. 125 km/h to mi/h
b. 225 km/h to ft/s
c. 2.70 kg/m3 to g/cm3
d. 2.70 g/cm3 to lb/ft3
18. A children‘s pain relief elixir contains 80. mg acetaminophen per 0.50 teaspoon.
The dosage recommended for a child who weighs between 24 and 35 lb is 1.5
teaspoons. What is the range of acetaminophen dosages, expressed in mg
acetaminophen per kg body weight, for children who weigh between 24 and 35 lb?
19. One metal object is a cube with edges of 3.00 cm and a mass of 140.4 g. A second
metal object is a sphere with radius of 1.42 cm and a mass of 61.6 g. Are these
objects made of the same or different metals if the calculated densities are accurate
to ±1.00%?
20. What is the speed of light?
21. How long is a light year?
22. Plot the following data points:
x 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
y 1.2 1.6 2.1 2.5 2.8 3.3 3.7 4.1
What type of relationship exists?
Predict the value of y when x 0.9
23. Plot a graph of y vs x
x 0.0 2.5 3.6 5.4 7.7 8.3 9.5 11.1
y 0.0 3.1 6.5 14.6 29.6 34.5 45.1 61.6
What type of relationship appears to exist?
Plot a graph of y vs x to confirm
24. Rearrange the following formulae to make the symbol in brackets the subject
s m
a. v (t) e. T 2 (m)
t k
b. F ma (a) KE 1 mv 2 (v)
f. 2
m
c. D (m) v2
v g. a (v)
r
v
d. a (t) h. Q mcT (m)
t
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GM 1 1 1
i. v (r) k. (E)
r P E S
T0
F
Gm1 m2 l. T (v)
1 v
j. (d) 2
d2 c
25. Determine the magnitude and argument of the following vectors:
a. a (3,4) e. e 2 a b
~ ~ ~ ~
b. b (1,3) f. f 6 c d
~ ~ ~ ~
c. c i 4 j g. g a d
~ ~ ~ ~ ~
~
d. d 5 i 5 j h. h 6 i b
~ ~ ~ ~ ~ ~
26. Add the following vectors together:
a. b.
27. Determine the average speeds for the following journeys. (Either km/h or m/s):
a. 20 km in 30 min
b. 150 m in 45 s
c. 789 cm in 3.4 s
d. 7500 km in 15.6 h
28. Determine the time taken to travel the following distances at the given speeds:
a. 150 km at 75 km/h
b. 250 m at 15 m/s
c. 2.56 km at 6 m/s
d. 400 m at 100 km/h
29. Determine the distances travelled at the following speeds
a. 100 km/h for 15 min
b. 56 m/s for 1 h
c. 120 km/h for 20 min
d. 4.56 m/s for 12 min
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30. John and Jack are having a race over 1500m. John is the faster runner (4.2m/s), so
he gives Jack (who can run at 3.8m/s) a 30sec head-start.
a. Who wins the race?
b. As the winner crosses the finish line, how far behind is the other runner?
31. An aeroplane flys at 750 km/h for one third of its flight time and averages 720 km/h
for the entire trip. What is the average speed, in km/h over the remaining part of the
journey?
32. A car accelerates uniformly from rest at 5.5m/s2 over a distance of 100m. What is its
final velocity?
33. What is the acceleration of a vehicle that accelerates uniformly from 12 m/s to 20.5
m/s over a distance of 15 m?
34. A car accelerates uniformly from 2m/s at 5m/s2 for 15s. How far does it travel
during this time?
35. A body falls 15m in 0.8s. What was its initial velocity?
36. How long does a particle take to accelerate uniformly from 2.5 m/s to 20.5 m/s at
3.5 m/s2?
37. A plane accelerates uniformly for 5.5 seconds at 5.5m/s2. If its final velocity is
150m/s, what was its initial velocity?
38. How far does a particle travel when accelerating from 2m/s to 15m/s at 5m/s2?
39. A plane accelerates uniformly over 200m at 8m/s2. If its final velocity is 500m/s,
what was its initial velocity?
40. What is Newton‘s First Law of motion?
41. What is Newton‘s Second Law of motion?
42. What is Newton‘s Third Law of motion?
43. Which of Newton‘s Laws of motion is about inertia?
44. Which of Newton‘s Laws of motion is about momentum?
45. Find the force required to push a 2 kg object through a distance of 2 m from rest in 5
s, assuming there is no friction.
46. Explain how the motion of a rocket taking off is an example of an action-reaction
situation.
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