SCIENCE 1206
Unit 4 - Physics
SIGNIFICANT DIGITS AND CALCULATIONS

Scientists generally work with two kinds of data
values:
1.
2.

Measure values – values taken from experiments. Also
called experimental values.
Calculated values – values determined from
mathematical calculations. Also called theoretical
values.
Base quantities (SI)







Length – meter (m)
Time – second (s)
Mass – gram (g)
Electric current – ampere (A)
Temperature – Kelvin (K)
Amount of a substance – mole (mol)
Luminous intensity – candela (cd)
MEASURED VALUES AND SIGNIFICANT
DIGITS
Depends on the calibration of the instrument
being used. For example, a typical metric ruler is
marked at 1 cm intervals along with unnumbered
calibration marks in between the numbered ones
(mm). These divided each 1 cm interval into 10
equally spaced segments. Each segment is equal
to 0.1 cm or 1 mm.
 These markings represent the instrument’s
standard for measuring length, therefore, any
length measured based upon these markings
must be considered significant.

MEASURED VALUES AND SIGNIFICANT
DIGITS
It is possible for the human eye to estimate
between each 0.1 cm (or 1.0 mm) markings but it
is just that, an estimation, and one person’s
estimation may be different from another’s.
 For example, Jessica might say that a piece of
wood is 9.66 cm long while Carly says that the
same piece of wood is 9.68 cm long.
 Notice that the first two numbers in each
person’s estimation is the same (9.6) but the last
is not. In this example, the length of wood has 3
significant digits.

MEASURED VALUES AND SIGNIFICANT
DIGITS

Ex. What would you say is the length if the
figure below?
CALCULATED VALUES AND SIGNIFICANT
DIGITS
Def’n: significant digits (a.k.a significant figures) –
includes all those digits that are certain, plus one
digit that is estimated (i.e. rounded).
Rules Regarding Significant Digits:
1. All non-zero digits (1 – 9) are significant
2. Rules for zeros:
a.
b.
Zeros between two non-zero digits are always
significant. For example, 5003 has 4 significant
digits
If a number contains a decimal and only a zero to
the left of that decimal then all zeros to the left of a
non-zero digit are not significant, i.e. they are just
place holders. For example, 0.0000408 has 3
significant digits.
CALCULATED VALUES AND SIGNIFICANT
DIGITS
c.
d.
e.
All zeros that lie to the left of a non-zero digit
of a number that does not contain a decimal
are not significant, i.e. they are just place
holders. For example, 3000 has 1 significant
digit.
If a number has a decimal point, all zeros to
the left of the decimal place are significant.
For example, 20.7 has 3 significant digits.
If a number has a decimal point, and ends in
a zero, then all digits are significant. For
example, 20.00 has 4 significant digits.
CALCULATED VALUES AND SIGNIFICANT
DIGITS

Ex. How many significant digits does each of the
following numbers have?

0.000406

120.00420

500000

10.68

0.124

3.67

2000.9
SCIENTIFIC NOTATION AND SIGNIFICANT
DIGITS

Two advantages to using scientific notation:
1.
2.
Tells you the number of significant digits directly
since non-significant digits are omitted.
Easier to express very large or very small numbers.
To write a number using scientific notation, we
need to know how many significant digits we
need.
 Scientific notation is written as a number times a
power of 10. The beginning number can only
have one digit before the decimal but can have
several numbers after the decimal. For example,
3.56 × 107 is a number in scientific notation.
 To determine the original number you will have
to expand the power and multiply.

SCIENTIFIC NOTATION AND SIGNIFICANT
DIGITS

1.
Ex. Write each number using scientific notation.
654 000 000
2.
0.000 000 025
3.
740 000 000 000
4.
0.000 002 45
SCIENTIFIC NOTATION AND SIGNIFICANT
DIGITS
Calculated Values:

The product or quotient will be written as many
significant digits as the number involved in the
operation with the least number of significant
digits.
Ex. 0.000460 x 200.80 =

Ex. (2.000 x 1021) ÷ (6.0 x 10 -13) =

SCIENTIFIC NOTATION AND SIGNIFICANT
DIGITS
Sometimes we have to express the answer in
scientific notation in order to get the correct number
of significant digits.
 Ex (6.0)(500.0) =


Since 6.0 has two significant digits and 500.0 has
four significant digits, the answer must have two but
3000 only has one and 3000.0 has five. Therefore, we
must use scientific notation to get
_____________________.
SCIENTIFIC NOTATION AND SIGNIFICANT
DIGITS
The sum or difference has the same number of digits
after the decimal as the number with the least
number of digits after the decimal.
 Ex 693.45 + 5.3 =


Ex 437.358 – 68.641 =
*Worksheet #1
ACCURACY AND PRECISION
Accuracy – indicates how close a measurement is
to the accepted value. For example, if a standard
100 g mass is placed on a balance, the balance
should read 100 grams. If the balance does not
read 100 g, then the balance is inaccurate.
 Calibration – the process of evaluating and
adjusting the precision and accuracy of
measuring equipment.
 Precision – indicates how close together or how
repeatable the measurements are. A precise
measuring instrument will give very nearly the
same measurement each time it is used.

ACCURACY AND PRECISION
ACCURACY AND PRECISION
Both accuracy and precision are important in
dealing with measurements.
 However, in some cases, it is far more important
for the measuring device to be precise than to be
accurate.


Consider an experiment in which the difference in
mass is being measured. Since one measurement of
mass will be subtracted from another, the error due
to any inaccuracy in both measurements will be
cancelled, provided both measurements were precise.
INSTRUMENTS USED IN EXPERIMENTS

What type(s) of instruments would you use to
measure the following:







Temperature
View very small things
View very far things
Time
Analyze data
Mass
Volume
SOURCES OF ERROR
Measurement always involves uncertainty or
random error (i.e. no instrument is perfect)
Measurement always involves uncertainty or
random error.
For example, the use of the cm ruler described before
will have a degree of uncertainty associated with
estimating the final significant digit.
It is good practice to record the precision or
estimated uncertainty as part of the
measurement.
For example, 4.50 ± 0.02 cm.
SOURCES OF ERROR
Discrepancy is when there is a difference
between the value determined by your
experimental procedure and the value that is
generally accepted or considered as average.
Systemic error is error due to the use of an
incorrectly calibrated measuring device.
Experimental error is the comparison of an
experimental value to an known value
(sometimes called the actual value or theoretical
value)
(experimental value) − (known value)
% error =
× 100
known value
AVERAGE SPEED
Def’n: average speed – the total distance divided by the total
time for a trip. The symbol for average speed is vave.
The formula for determining average speed is:
∆𝑑
𝑣ave =
∆𝑡
∆ means “change in” (Greek letter delta), therefore, ∆d means
“change in distance” and ∆t means “change in time”
AVERAGE SPEED

To calculate ∆d and ∆t you need two points. In other
words,
∆d = d2 – d1

and
∆t = t2 – t1
d1 and t1 are usually (but not always) the initial distance
and time, respectively, which are zeroes.
QUANTITY SYMBOLS
When writing quantity symbols (letters used to represent
quantities such as distance, time, and speed) are written
in italics. Unit symbols are written normally.
 For example, m is the quantity symbol for mass while m
is the unit symbol for metres.

TYPES OF SPEEDS
1.
Instantaneous Speed



The speed at which an object is travelling at a
particular instant.
For example, the speedometer on a car gives the
driver the instantaneous speed of the car at any
given time.
An object’s instantaneous speed is not effected by its
previous speeds or by how long it has been
travelling.
TYPES OF SPEEDS
2.
Constant Speed




A.k.a uniform motion
When an object has the same instantaneous speed
over a period of time we say that it is constant.
If you were to graph a distance vs. time graph of an
object in uniform motion, then the graph would be
linear.
Constant speed is actually fairly uncommon because
of friction (the resistance caused by two things
moving against each other) and gravity which
causes the thing(s) to slow down
CALCULATING AVERAGE SPEED
Ex 1. A train travels 386 km in 3.0 hours. What is
the trains average speed in km/h?
Ex 2. A child runs 200 meters in 40 seconds. What
is the child’s average speed in m/s?
CALCULATING AVERAGE SPEED
Ex 3. A worm travels 40 m in 20 mins. What is
the worms’ average speed in m/s?
Ex 4. How far would you travel if you drove at 115
km/h for 45 mins?
CALCULATING AVERAGE SPEED
Sometimes uniform motion occurs in two parts.
This means that part of the motion occurs at one
speed and the rest occurs at another.
 To calculate the average speed for these
situations you DO NOT calculate the speed for
the first part, calculate the speed for the second
part and then average the two speeds!
 To calculate the average speed in these situations
you take the total distance covered and divide it
by the total time.

CALCULATING AVERAGE SPEED
Ex. A traveler journeys by plane at 400.0 km/h for
5.0 hours and then by car at 100.0 km/h for 2.0
hours. What is her average speed?
CALCULATING AVERAGE SPEED
Ex. You travel 30.0 km in motorboat at 6.0 km/h
and then 20.0 km on a sailboat at 10.0 km/h. What
is your average speed?
CALCULATING AVERAGE SPEED
Your turn.
Page 358 – 359 #1 – 4, 7(tricky!!) – 8, 10 – 11
(Don’t forget to wipe the dust off your textbooks
first)
DISTANCE – TIME GRAPHS
Just as in math, time is the independent variable (x –
axis) and distance is the dependent variable (y – axis).
 Recall that the equation of a line is y = mx + b where y is
the dependent variable, x is the independent variable, m
is the rate of change and b is the y – intercept.
 In the case of an object with constant speed, the equation
becomes ∆d = v∆t + 0 since d is the dependent variable,
t is the independent variable, v is the rate of change since
v = ∆d/∆t which is essentially the rise/run and since the
initial time is 0, the y-intercept will be 0.

DISTANCE – TIME GRAPHS
Q. What happens if the y – intercept is not 0? What does it
mean in terms of the object’s distance and time?
A. If the y – intercept is not zero then this means that the
object started somewhere other than the starting point.
I.e. The object was so far (whatever value the yintercept is) from the starting point.
DISTANCE – TIME GRAPHS

Textbook page 363
Plotting a distance – time graph, you plot the points
given the same as you do in math.
 Determining the rate of change (i.e. speed) from a
distance – time graph, is the same as in math. I.e.
rise/run or (y2 – y1)/(x2 – x1)
 The steeper the line, the faster the object is moving.

DISTANCE – TIME GRAPHS
Graphs give us a visual.
 Recall that the dependent variable is plotted on
the y axis while the independent variable is
plotted on the x axis.
 Always label your axis and title you graph.
(Doesn’t need to be anything fancy, just what
they represent).
 Always use a proper scale (i.e. go up by the same
amount each time). You want to chose a scale
that allows you to plot your points as easily as
possible and makes your graph a reasonable size.

DISTANCE – TIME GRAPHS

Recall that when we have points that do not line
up perfectly but have a linear trend, we draw a
line of best fit.
DISTANCE – TIME GRAPHS

If the points form a curve, then draw a curve
through the points.
DISTANCE – TIME GRAPHS
Distance is always plotted on the y axis and time
is always plotted on the x axis
 The steepness of the line, which is known as the
slope of the line, is equal to the speed of the
object.


Slope and rate of change are the same thing.
Therefore, the slope of a line is the rise over the run
𝑦 −𝑦
or 2 1
𝑥2 −𝑥1

If you have a broken line graph, then to calculate
the average speed you use the initial point and
the final point.
DISTANCE – TIME GRAPHS
Ex. Emily is biking from Corner Brook to St.
John’s. The table below displays her distance every
12 hours. Time (hours)
Distance (km)
0
0
12
180
24
360
36
540
48
720
a. Draw a properly labeled distance – time graph.
DISTANCE – TIME GRAPHS
b.
What is Emily’s average speed?
DISTANCE – TIME GRAPHS
Ex. The data in table was collected for an object
travelling at a uniform speed.
Time (s)
Distance from First
Marker (m)
0.0
0.0
1.0
8.0
2.0
16.0
3.0
24.0
4.0
32.0
5.0
40.0
a. Use the data to make a graph.
b. Use the graph to determine the slope of the line.
c. What value does the slope of the line represent?
d. Describe a situation in which such data may have
been collected.
DISTANCE – TIME GRAPHS
Quite often, a d-t graph will be a broken line
graph, with each section representing a different
part of the journey.
 When a line segment has a positive slope, this
means that the object is moving forward.
 When a line segment has a slope of 0 (horizontal
line), this mean that the object is stopped.
 Remember to calculate the average speed on
these types of graphs, you join the end point with
the initial point and find the slope of this line.

DISTANCE – TIME GRAPHS.
Ex. Calculate the average speed for each object in
each graph below.
a. y
30
20
10
5
10
15
x
DISTANCE – TIME
d
b.
60
40
20
10
20
30
*Practice Problems page 365 #2, 3, 5, 6
t
SPEED – TIME GRAPHS (UNIFORM
MOTION)
We will only be looking at graphs of objects with
uniform motion right now.
 If we are to plot any graph of an object with
uniform motion on a speed – time graph, what
shape will the graph take?



Since the speed does not change over time, the graph
will be a horizontal line.
What does the y-intercept represent in a speed –
time graph?

The starting and, hence, the constant speed of the
object
SPEED – TIME GRAPHS (UNIFORM
MOTION)

What does the area under the graph (i.e. the line)
represent?

The distance travelled (recall that ∆d = ∆v∙t and the
areas under the line is a reactangle with area =
length x width)
SPEED – TIME GRAPHS (UNIFORM
MOTION)
Ex. What is the speed and the distance travelled in
the graphs below?
a.
s
60
40
20
10
20
30
t
SPEED – TIME GRAPHS (UNIFORM
MOTION)
s
100
90
80
70
60
50
40
30
20
10
10
20
30
40
50
t
SCALAR VS. VECTOR QUANTITIES
A scalar quantity is a quantity that only has size.
For example, time and mass are scalar
quantities.
 A vector quantity is a quantity that not only has
size but has direction too. For example,
temperature and height above sea level are
vector quantities.
 Vector quantities are represented by symbols
that include a small arrow over the quantity
symbol while scalar quantities symbols do not
have arrows. For example, 𝑣 is the symbol for
velocity while 𝑣 is the symbol for speed.
 To check to see is a quantity is a scalar or a
vector see if you can negative numbers; if you
can’t then it is a scalar; if you can then it is a

DISTANCE VS. DISPLACEMENT
Distance tells us how far someone or something
has travelled in total. It does not tell which
direction that someone or something travelled
nor where they/it ended up in relation to their
starting position.
 In order to define displacement, we need to first
know what is meant by position. An object’s
position is how far and in what direction it is
from a reference point. Depending on what is
chosen for the reference point, the position may
change even though the object has not moved. In
other words, position is not a fixed value. The
symbol for position is 𝑑.

DISTANCE VS. DISPLACEMENT
Therefore, displacement is the change in position,
i.e. where the moved in relation to its starting
point or to some other fixed point.
 Because displacement is a vector quantity, i.e.
shows direction, its symbol will have an arrow
above it and since it is the change in position the
symbol for displacement is ∆𝑑.

DISTANCE VS. DISPLACEMENT

Since displacement is the change in position, in
order to calculate displacement you need to
subtract the first position from the second
position:
∆𝑑 = 𝑑2 − 𝑑1

Remember that displacement is a vector quantity
so you may and can have negative answers!
DISTANCE VS. DISPLACEMENT

One way to look at distance and displacement is
to consider the number line. Look at the line
below. The top arrow indicates that a person
started at the zero position, moved 3.0 km East
(or to the right), and then moved 4.0 km West (or
to the left).
Q. What is the distance?
 A. The distance is the complete path covered,
which is 3.0 km + 4.0 km = 7.0 km.

DISTANCE AND DISPLACEMENT
Q. What is the displacement?
 A. The displacement is the change in the
person’s position, which is -1.0 km because that is
the person’s final position on the number line.
 Notice that the plus symbol is used for a
displacement to the right, East, or North, and a
negative symbol is used for displacement to the
left, West or South.
 Note: that when calculating just the distance it is
addition, without signs, while when calculating
speed is it the change in distance which is
subtraction.

DISTANCE AND DISPLACEMENT
Confusion Warning!
When you are calculating the change in position
(i.e. the displacement) of ONE move, then you
subtract the final position from the initial
position as it is the change you are after, i.e. how
far and in what direction.
When you are calculating the change in position
(i.e. the displacement) of more than one move,
then you add all the changes of position as it is
the total displacement you are after, i.e. where
the object ends up after all the changes in
position. This is known as the resultant
displacement.
DISTANCE AND DISPLACEMENT

Ex. When cross country skiing one afternoon you
left the lodge and first skied 8 km north on the
Bunny Trail. At this point you decided to branch
off and ski to the Wilderness Hut. You had to ski
10 km north on the Bear Trail to get to it. At the
hut what is your displacement with respect to the
lodge?
DISTANCE AND DISPLACMENT
Ex. A car travels 10 km [N] then turns and goes 8
km [S]. What is the cars distance and
displacement with respect to its initial position?
DISTANCE AND DISPLACEMENT
Ex. Susan runs 10 times around a circular track
with a radius of 100 m. What is Susan’s total
distance travelled and what is her displacement
with respect to the starting line.
DISTANCE AND DISPLACEMENT


Read pages 414 – 415 in textbook. Questions
page 416 – 417 #1,2, 5, 6, 13
Questions page 423 # 5a, 6, 7
VELOCITY

Recall that speed is the rate of change of
distance.
Velocity is a speed in a particular direction. For
example, 30 km/h is a speed but 30 km/h [N] is a
velocity.
 Hence, velocity is the rate of change of
displacement.
 Just like speed, we can only calculate the average
velocity.

VELOCITY
Recall that quantities which have magnitude
(size) but no direction are scalars while
quantities which have magnitude (size) and
direction are vectors.
 Hence, velocity is a vector quantity while speed is
a scalar quantity
 Therefore, the equation for average velocity will
contain arrows over the vector quantities. Note
that time is a scalar quantity in any equation so
it never has an arrow about its symbol.
∆𝑑
𝑣𝑎𝑣𝑒 =
∆𝑡

VELOCITY

Some textbooks uses bold type for vectors and
italics for scalars so be careful!! I will use arrows
for vector symbols and no arrows for scalar
symbols since this is how you have to write them
on paper.
VELOCITY
Ex 1. Ants on a picnic table walk 130 cm to the
right and then 290 cm to the left in a total of 40 s.
Determine the ant’s distance covered, displacement
from original point, average speed, and average
velocity.
VELOCITY

What these number mean is:
420 cm is the total distance covered by the ants
 -160 cm means that the ants ended up 160 cm to the
left of their starting position
 10.5 cm/s is the average speed that the ants were
travelling for the entire trip
 -4.0 cm/s means that the ants were, on average,
travelling to the left 4.0 cm/s

VELOCITY
Ex 2. What is the velocity of a car that travelled a
total of 75 km north in 1.5 hours?
VELOCITY
Ex 3. Carly walks 1.2 km [W] and then walks 2.4
km [E] in 1.5 hours. What is her average speed
and her average velocity?
VELOCITY
Ex 4. A car travels 250 km [N] in 2.5 hours and
then travels 175 km [E] in 1.5 hours. What is the
cars average speed and average velocity?
VELOCITY
Ex 5. Keisha is pedaling her bicycle at a velocity of
0.10 km/min [S]. How far will she travel in two
hours?
VELOCITY
Questions page 436 #1, 5, 6
**Don’t forget to write the formulas!!!

POSITION – TIME GRAPHS

A position – time graph shows an objects change
in position over time.
These are similar to distance – time graphs.
 The main difference is that position – time graphs
can contain line segments with negative slope and
can have negative y-intercepts.
 Recall that negatives in these situations only
indicates the position of an object in relation to a
specified point, i.e. an object starting to the left of the
reference point.

POSITION – TIME GRAPH
A positive slope on a p-t graph means the object
has a fixed speed to the right (or similar
direction)
 A negative slope on a p-t graph means the object
has a fixed speed to the left (or similar direction)
 A slope of 0 (i.e. a horizontal line segment), still
means that an object has stopped on a p-t graph

POSITION – TIME GRAPHS
A positive y-intercept means the object started to
the right of the reference point.
 A negative y-intercept means the object started
to the left of the reference point.
 A y-intercept at the origin means that the object
started at the reference point.

POSITION – TIME GRAPHS
Ex. What would the following position – time
graphs represent?
POSITION – TIME GRAPHS
Position (m) [N]
Ex. Use the graph below to answer the questions.
a. Where did the object
start?
of
b. What is the position
the object at 3.0 s?
c. What is the velocity of
the object at 9 s?
POSITION – TIME GRAPHS
d.
When was the object moving north?
e.
When was the object moving south?
f.
When was the object stopped?
POSITION – TIME GRAPHS

A position time graph which shows non-uniform
motion (i.e. acceleration) would be displayed as a
curved line on a graph.

To determine the velocity of an object at a certain
point in time (i.e. instantaneous velocity), we would
need to determine the slope of the tangent to the
point on the graph.


A tangent is a line that is drawn along a curve that only
touches the curve at one point.
Tangents may be either positive or negative.
POSITION – TIME GRAPHS
Position (m) [N]
Ex. Determine the velocity at 20 s on the graph below.
Time (s)
POSITION – TIME GRAPHS

Questions page 450 – 451 #1, 2, 3, 4, 5, 6, 7
ACCELERATION
Def’n: acceleration – the rate of change of velocity
per unit of time, i.e. how much an object is
speeding up or slowing down.
 Since velocity is a vector quantity, acceleration is
also a vector quantity and, hence, has direction
associated with it.
 Since velocity units are m/s, or likewise, and we
are dividing by the time, the unit for acceleration
will be m/s2, or likewise.

aave

v

t
ACCELERATION
Since acceleration is the rate of change of velocity
over time, the equation is often written as:
𝑣𝑓 − 𝑣𝑖
𝑎=
𝑡
 This version of the formula is used because often
objects accelerate while the are already moving,
therefore, the initial velocity is not always 0.
 A positive acceleration means the object is
speeding up while a negative acceleration means
that the object is slowing down.

ACCELERATION
Ex. A skater increases her velocity from 2.0 m/s to
10.0 m/s in 3.0 seconds. What is the skater’s
acceleration?
ACCELERATION
Ex. A car accelerates at a rate of 3.0 m/s2. If its
original speed is 8.0 m/s, how many seconds will it
take the car to reach a final speed of 25.0 m/s?
ACCELERATION
Ex. While travelling along a highway, a driver
slows from 24 m/s to 15 m/s in 12 seconds. What is
the car’s acceleration?
ACCELERATION
Ex. A cart rolling down an incline for 5.0 seconds
has an acceleration of 4.0 m/s2. If the cart has a
beginning speed of 2.0 m/s, what is its final speed?
*worksheet #3
VELOCITY – TIME GRAPHS
On a velocity time graph, a horizontal line
represents a constant velocity (i.e. uniform
motion) and an oblique line represents a
changing velocity (i.e. non-uniform motion).
 A horizontal line above the t-axis represents a
uniform motion in positive direction.
 A horizontal line below the t-axis represents a
uniform motion in negative direction.

VELOCITY – TIME GRAPHS
VELOCITY – TIME GRAPHS
VELOCITY – TIME GRAPH

Much like with speed – time graphs, we can find
the displacement from a velocity – time graph by
finding the _______________________________.
VELOCITY – TIME GRAPHS
Velocity (m/s) [N]
Ex. Determine the displacement from the graph
below.
Time (s)
VELOCITY – TIME GRAPHS

We can also construct a velocity – time graph
when we are given a displacement – time graph.
To do this, we need to determine the velocity of each
line segment on the displacement – time graph.
 These graphs look a bit “strange” as the will be
horizontal lines that seem to “jump” all over the
place.

VELOCITY – TIME GRAPHS
Velocity (m/s) [E]
Position (m) [E]
Ex. Use the 𝑑 − 𝑡 graph below to construct a 𝑣 − 𝑡
graph.
ACCELERATION ON GRAPHS

On a distance – time graph or a displacement –
time graph, acceleration would be represented by
a curved line.
d (m)
10
8
6
4
2
2
4
6
8
10
t (s)
ACCELERATION ON GRAPHS

On a speed – time or a velocity – time graph,
acceleration is represented by an oblique line if
the acceleration is constant.
ACCELERATION ON GRAPHS

On a speed – time or velocity – time graph,
acceleration is represented by a curve if the
acceleration is not constant.
v (m/s)
v (m/s)
10
10
8
8
6
6
4
4
2
2
2
4
6
8
10
t (s)
2
4
6
8
10
t (s)
ACCELERATION ON GRAPHS

To determine the acceleration on a curve at a
certain point in time, you need to find the slope of
the tangent drawn at that point.
VELOCITY – TIME GRAPHS
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