Science 10 Motion. Numbers vs. Measurement There is a difference in between numbers used in math and...

Science 10

Motion

Numbers vs. Measurement

• There is a difference in between numbers used in math and measurement used in science.

• In math, every number carries importance• In science, not every number in a

measurement carries the same importance.– More important numbers are called significant figures.– Less important numbers are called place holders.

Measurement

• Every measurement contains an exact amount of significant figures.– It includes all numbers that were measured

from the scale used.– Plus 1 extra ‘guessed number’ that is not on

the scale.

• Always include one more number than your scale tells you!

0cm 100cm

Scale does not tell us any certain numbers, so we can only write down 1 guessed number

50

4 is a certain number and it is significant

0 is a place holder and is not significant.

The scale now tells us the tens so we can be certain of those numbers. We add a guessed number after the ones we are certain of.

488 is a guessed number and it is significant

5 is a guessed number and it is significant

Our scale tells us the tens and the ones so we can be certain of those numbers. We add a guessed number after the ones we are certain of.

47.0

47 are certain numbers and are significant

0 is a guessed number and it is significant

Rules for Significant Figures

• There are 2 rules for determining the number of significant figures.

1. Decimal rule- (use this rule when the measurement contains a decimal)

– Count the numbers from left to right beginning at the first non-zero number.

0.001234-

1.234-

0.123400-

12340.0-

12340.-

1.2340 x 10-3-

4 sig. figs.

4 sig. figs.

6 sig. figs.

6 sig. figs.

5 sig. figs.

5 sig. figs.

Rules for Significant Figures

2. Non-decimal rule- (use this rule when the measurement does not contain a decimal)

– Count the numbers from right to left beginning at the first non-zero number.

1234-

102340-

0.123400-

12340-

12340.-

100002-

4 sig. figs.

5 sig. figs.

6 sig. figs.

4 sig. figs.

5 sig. figs.

6 sig. figs.

Scientific Notation

• Scientific notation is a method of writing numbers that:– Can make large numbers more easy to read.– Indicate the proper number of significant

figures.

Rules for Writing in Scientific Notation

1. Write down all the significant numbers2. Put a decimal after the first number. (the

number will now be between 1-10)3. Write “x 10”4. Write the power corresponding to the number

of places the decimal was (would have) been moved. (Moving right is negative, moving left is positive)

• Count the number of digits between where the decimal was before and where it is now

25 000 000 000 000

1.Write down all the significant numbers

25

2.Put a decimal after the first number. (the number will now be between 1-10)

.

3.Write “x 10”

x 10

4.Write the power corresponding to the number of places the decimal was (would have) been moved. (Moving right is negative, moving left is positive)

13

0.000 000 000 030 0

1.Write down all the significant numbers

300

2.Put a decimal after the first number. (the number will now be between 1-10)

.

3.Write “x 10”

x 10

4.Write the power corresponding to the number of places the decimal was (would have) been moved. (Moving right is negative, moving left is positive)

-11

How do you write the number 10 000 with 3 significant figures?

100. x 104

Change 0.00123 x 10-3 into proper scientific notation.

123. x 10

-3 -3= -6

-6

Calculating using Significant Figures

• There are 2 rules for calculating with significant figures.

1. Precision rule- (used for addition and subtraction)

– The answer will have the same precision as the least precise measurement from the question.

10 cm

10. cm

10.0 cm

10.00 cm

Least precise

Most precise

1.234+0.05678

1.29078

This value is the least precise value. The answer will end at the same spot.

Round the value after the last sig. fig.

=1.291

12340+5678000

5690340

This value is the least precise value. The answer will end at the same spot.

Round the value after the last sig. fig.

=5.690 x 106

=5690000

Calculating using Significant Figures

2. Certainty rule- (used for multiplication and division)

– The answer will have the same number of significant figures as the least number of significant figures from the question.

123X 45

3 significant figures

2 significant figures

The answer will have 2 significant figures

55351 2

Round the value after the last significant figure

5500Place holders

450X 0.0123

2 significant figures

3 significant figures

The answer will have 2 significant figures

5.5351 2

Round the value after the last significant figure

5.5

Units

• A unit is added to every measurement to describe the measurement.Ex.– 100 cm describes a measured length. – 65 L describes a measured volume.– 12.4 hours describes a measured time.– 0.011 kg describes a measured mass.

Units

• In Canada we use the metric (SI) system.

• The metric (SI) is a system designed to keep numbers small by converting to similar units by factors of 10.

• Prefixes are added in front of a base unit to describe how many factors of 10 the unit has changed.

Units

• Base units of measurement are generally described by one lettre.– m- metre (length)– s- second (time)– g- gram (mass) *The base unit for mass is

actually the kg (kilogram)– L- litre (volume)

Units• Prefixes

• Prefixes are added to the front of any base unit.Ex. mm, cm, dm, m, dam, hm, km

Kilo hecto deca base deci centi milli (k) (h) (da) (d) (c) (m)

Converting units

• There are 2 methods to convert units1. Step Method- count the number of places to

move the decimal.

2. Dimensional Analysis- multiplication by equivalent fractions of 1.

Converting Units

• Step method-– Move the decimal the same number of spaces

and direction as the distance in between prefixes.

Kilo hecto deca base deci centi milli (k) (h) (da) (d) (c) (m)

Convert 34.56 cm into m

It starts at centi (for centimetre)We move 2 spaces to the left to get from centimetre to metreSo we move the decimal 2 places to the left in our number

3456.0 cmm

Kilo hecto deca base deci centi milli (k) (h) (da) (d) (c) (m)

Convert 21.0 kg into g

It starts at kilo (for kilogram)We move 3 spaces to the right to get from kilogram to gramSo we move the decimal 3 places to the right in our number

210. kgg00 =2.10 x 104g

Converting Units

• Dimensional Analysis-– Multiply the measurement by a fraction that

equals 1– The fraction will contain the old unit and the

new unit.– The fraction must cancel out the old unit.

(follow the rule that tops and bottoms cancel out)

Convert 34.56 cm into m

34.56 cm

Multiply by a fractionThe fraction must contain the new and old unit. Tops and bottoms cancel out

cm

m

We need to make the fraction equal 1. Put the larger measurement as 1 and add 0’s to the smaller measurement. (# of zeroes equals the number of places the prefixes are moved.

1

Kilo hecto deca base deci centi milli (k) (h) (da) (d) (c) (m)

Move 2 places so we need 2 0’s

100

Multiply the tops and divide the bottoms

= 0.3456 m

Counted and exact values do not count for significant figures

Convert 21.0 kg into g

21.0 kgkg

g

1

1000= 21000g= 2.10 x104g

Convert 15.0 m/s into km/h

15.0 m s

We follow the same rules, but we convert 1 unit at a time.

m

km

min

s

h

min1

1000

60

1

60

1

We multiply the tops, and divide the bottoms.

= 54.0 km/h

Convert 80.0 km/h into m/s

80.0 km h

We follow the same rules, but we convert 1 unit at a time.

mkm

hs1

10003600

1

We multiply the tops, and divide the bottoms.

= 22.2 m/s

Defined Equations

• Relationships between variables can be expressed using words, pictures, graphs or mathematical equations.– A defined equations is a mathematical

expression of the relationship between variables

Ex. Mass and Energy are related by the speed of light

E = mc2

Defined Equations

• Defined equations can be manipulated to solve for any of the variables.– We use the same principles from math.

• There are 2 rules that must be followed to isolate a variable.1.It must be alone

2.It must be on top (numerator)

Solve E = mc2 for m

m must be isolated

E = mc2

m is already on top so we will not touch m. We have to isolate m by moving c2 to the other side.

Divide both sides by c2

c2c2E = m

Solve d = m/v for v

d = mv

v is on the bottom so we need to move v first and then isolate.

Multiply by v on both sides

v vDivide by d on both sides

d d

Speed

• The distance travelled by the amount of time.– How fast something is moving.

v = Δd

Δt

• Speed is measured in m

s

Speed

• You can look at speed in 3 different ways– Average- the speed over the whole trip.

• Total distance divided by total time.

– Instantaneous- the speed at one point in the trip.

• Looking at the speedometer.

– Constant- the speed remains the same over a period of time.(uniform motion)

• Cruise control.

Calculations for speed

• Using the formula, v = d/t, we can make some mathematical calculations about speed.– Follow the same 3 steps to solve every

problem.1. Identify your givens and unknowns.

2. Identify the defined equation and isolate for the unknown variable.

3. Solve the equation using proper significant figures and units.

A trip to Calgary is 758 km. If you were to complete the trip in 7.25 h, what was you speed?

Givens

d= 758 kmt= 7.25 hv= ?

Formula

v = d t

Solve

v = 758km 7.25 h

v = 105km h

What type of speed did we calculate in the previous problem?

Average speed

If someone is travelling at a constant speed of 40.0 km/h, how far would they travel in 32.4 min.

Givens

d= ?t= 32.4 min

v= 40.0 km h

Formula

v = d t

Solve

d = 40.0km (0.54 h) h

d = 21.6 km

d = v t

32.4 min 1 h

60 min= 0.540 h

Representing Speed Graphically

• We can represent speed with words (fast, slow), numbers (32 km/h) and we can also represent it visually with a graph.– Speed is represented on a distance vs. time

graph.• The slope of the graph is the speed.

Time (s)

The slope of the line is equal to the speed

The steeper the slope, the greater the speed.A straight line indicates a constant speed.

Travelled the greatest distance in the same time. (fastest speed)

Travelled the least distance in the same time. (slowest speed)

A curved line indicates non-constant speed. (speeding up or slowing down.)

Time (s)

What is the speed of this graph?0 m/s

Where is the person going in this graph?

Back to the original starting position.

Describe the motion in the following graph? 1.Moving slowly at a constant speed

2.Moving faster at a constant speed

3.Not moving

4.Moving back to the start at a constant speed

5.Speeding up

Time (s)

Identify the 3 types of speed on the graph?Average Instantaneous Constant

the speed over the whole trip

the speed at one point in the trip

the speed remains the same over a period of time

Acceleration

• The change in speed by the amount of time. – How quickly something is speeding up (or

slowing down)

a = Δv

Δt

• Acceleration is measured in m

s2

Acceleration

• You can look at 2 types of acceleration.– Average- the acceleration over the whole

time period.• The change in speed over time.

– Constant- the acceleration remains the same over a long period of time.

Calculations for acceleration

• Using the formula, a = Δv/Δt, we can make some mathematical calculations about acceleration.– ‘Δ’ means change, Δv means change in

speed

– Δv = vfinal – vinitial

Or– Δ v = v2 – v1

A person on their bike changes their speed from 10.0 m/s to 15.0 m/s in 15.2 s. What is the acceleration of the bike?

Givens

Δ v=15.0m/s –10.0m/s = 5.0 m/sΔ t= 15.2 sa= ?

Formula

a = Δv Δt

Solve

a = 5.0m/s 15.2 s

a = 0.33m s2

A car is traveling down the road when they see an obstruction. The person accelerates at -3.2 m/s2 for 5.0 s until they stop. How fast was the car moving?

Givens

v1= ?v2= 0 m/sΔ t= 5.0 sa= -3.2 m/s2

Formula

a = v2- v1

Δt

Solve

v1=0 m/s –(-3.2m/s2)5.0s

v1 = 16 m/sv1 = v2- aΔt

Representing Acceleration Graphically

• We can represent acceleration with words (speeding up, slowing down), numbers (9.8 m/s2) and we can also represent it visually with a graph.– Acceleration is represented on a speed vs.

time graph.• The slope of the graph is the acceleration.• The area under the graph represents the distance

travelled

Time (s)

The slope of the line is equal to the acceleration

The steeper the slope, the greater the acceleration.A straight line indicates a constant acceleration.

Increased the speed the most in the same time. (fastest acceleration)

Increased the speed the least in the same time. (slowest acceleration)

A curved line indicates non-constant acceleration. (speeding up or slowing down at a changing rate.)

Time (s)

What is the acceleration of this graph? 0 m/s2, but it is a constant speed.

What is the acceleration of this graph?

A negative acceleration, they are slowing down.

Describe the motion in the following graph? 1.Constantly speeding up slowly

2.Constantly speeding up faster

3.Moving at a constant speed

4. Slowing down to a stop

Time (s)

Identify the 3 types of acceleration on the graph?Average Non-uniform Constant(uniform)

the change in speed over the whole trip

the speed is increasing at a changing rate

the change in speed remains the same over a period of time

Answer the questions using the following graph.

Time (s)10 20 30 50 60 70 80 90

2

4

6

8

10

12

14

16

18 How long did it take for the turtle to reach 14m?

It would take 70 sec.

How far did the turtle get in 25s?

It would make it about 1m

What is the average speed of the turtle over the entire trip?

v = d / t

v = 16.5m 78sv = 0.21m s

Answer the questions using the following graph.

Time (s)10 20 30 50 60 70 80 90

2

4

6

8

10

12

14

16

18 How long did it take for the hare to reach 10m/s?

It would take 58 sec.

How fast was the hare going at the 72s mark?

It would be travelling about 14.5 m/s

What is the average acceleration of the hare over the entire trip? a = v / t

a = 16.5m/s 78s

a = 0.21m s2

How far did the hare travel?

Distance is calculated using the area under the graph

A = ½ bh

b = 80 sH = 16 m/s

A = ½ (80s)(16m/s)

A = 640m