Mathematics Review
About Lecture Notes
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Note: Limitations in HTML make it difficult
to make readable mathematical equations.
Units:
All physical quantities have units.
"Units" are the scale of measure being used.
A time interval of "5" is meaningless unless you specify the units
(months, hours, seconds, or whatever).
Always include units.
Metric System:
We will use the metric (or SI) system in this course!
Most of you are probably more familiar with U.S. units, so why are we using
the metric system?
1. Because it is an easier and better system than the U.S. system.
2. Because the metric system is used in every country in the world
except one - the United States.
3. Because if you don't know it, it's about time you did!
Much of what we are doing today is summarized in the Appendix (pages
541-543) and Preview section P.9.
Base quantities in the metric system are
length: meter (m)
mass: kilogram (kg)
time: second (s)
These particular units are not what
make the metric system so good.
The real strength of the metric system is that a standard set of prefixes
is used to create larger and smaller units.
And these prefixes all change the units in simple multiples of 10 (or 100
or 1000).
Metric Prefixes:
| Prefix | Abbr. | Factor | Example |
| kilo | k | 10^3 = 1000 | 1 kg = 1000 g |
| mega | M | 10^6 = 1,000,000 | 1 Ms = 10^6 s |
| giga | G | 10^9 (a billion) | giga will not be used much |
| tera | T | 10^12 | tera will not be used much |
| centi | c | 10^-2 = .01 = 1/100 | 1 cm = .01 m |
| milli | m | 10^-3 = .001 = 1/1000 | 1 ms = 10^-3 s = .001 s |
| micro | (mu)* | 10^-6 | *this is the Greek letter mu |
| nano | n | 10^-9 | |
| pico | p | 10^-12 | pico will not be used much |
(You may be asked on a test what a particular metric prefix means or you may need to know what a prefix means in order to solve a math problem.)
Scientific Notation:
Scientific notation is a convenient system for working with very large (and
very small) numbers.
It is sometimes called "powers-of-ten" notation.
10^6 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000
10^3 = 1000
5 x 10^2 = 5 x 10 x 10 = 500
2.3 x 10^1 = 23
10^0 = 1
10^-1 = 1/10 = 0.1
6 x 10^-2 = 6 x (1/10) x (1/10) = 0.06
10^-3 = 1/(10^3) = 0.001
1. Write out 4.48 x 10^-2 in decimal form. _________
2. Simplify (4 x 10^2) / (2 x 10^-3) = ________
3. Simplify (6 x 10^-4) / [(10^3) (36 x 10^-10)] = ________
Basic Math Review:
Here a, b, and c represent
known numbers.
x is the unknown value we are trying to determine.
if ax = b, then x
= b/a
Proof: divide both sides of the equation by a
Example:
if (v)(4 s) = 10 m, then v = (10 m)/(4s) = 2.5 m/s
___________
if (1/x) = a, then x = 1/a
if 1/x = a/b, then x = b/a
if x/a = b/c, then x = (b/c)a = ab/c
if a/x = b/c, then xb
= ac (cross-multiplying)
and so x = ac/b
Other useful rules:
1/(1/a) = a
1/(b/a) = a/b
(a/b)/(c/d) = (ad)/(cb)
Angles:
360· = complete circle
90· is a right angle.
1· = 1/360 of a complete circle
1' = 1 minute of arc = 1 arcmin = 1/60·
1" = 1 second of arc = 1 arcsec = 1/60' = 1/3600·
Conversion factors: 1· = 60' 1' = 60"
Now do these practice problems:
4. Solve for x if x/(4 hr) = (20 dollars)/(8 hr).
5. Solve for x if (360·)/x
= (45·)/(100 km)
Answers
Conversion of Units:
Conversion factors: 1 ft = 30.48
cm = .3048 m, 1 in = 2.54 cm, 1 km = 1000 m, etc.
Many more conversion factors can be found in textbook appendices.
Example: How many inches is 10 cm?
Answer: 10 cm = 10 cm (1 in/2.54 cm) = 3.94 in
The term in brackets is just a factor of 1 because 1 in = 2.54 cm.
Note how the cm units cancelled out.
I recommend carefully following this procedure for all unit conversions.
Example: How many m^2 is 900 ft^2?
(both m^2
and ft^2
are units of area)
Answer: 900 ft^2 = 900 ft^2 (.3048 m/1 ft) (.3048 m/1 ft) = 83.6 m^2
Example: How many seconds in one year?
Answer: 1 yr = 1 yr (365.25 days/yr) (24 hr/day) (60 min/hr) (60
s/min) = 3.156 x 10^7 s
Note you can only convert between the same types of units.
It does not make sense to ask how many seconds in a kilogram.
Example: What is 60 mi/hr in m/s?
Answer: 60 mi/hr = 60 mi/hr (1 hr/3600 s) (1.609 km/mi) (1000 m/1
km) = 26.82 m/s
Do the following practice problems:
6. Convert 25.64 g to kg ________
7. Convert 9.8 m/s^2 to ft/s^2 ________
8. Convert 5 x 10^9 in^2 to ft^2 ________
9. Convert 19.3 g/cm^3 to kg/m^3 ________
Answers
Rate Problems: (called distance, velocity, time (D,v,t) problems in text)
Rate = Amount/Time
Given the Amount and Time, you can solve for the Rate.
Or, if given Rate and Time, you can solve for the Amount:
Amount = Rate x Time
Or, given Rate and Amount, you can solve for Time,
Time = Amount/Rate
Example: You are earning 25 cents per minute (25¢/min), how
much will you earn in an hour?
Solution: Rate = 25¢/min Time = 1 hr Amount = ?
Amount = Rate x Time = 25¢/min (1 hr) (60 min/hr) ($1/100¢) =
$15
[Or this could be done as a units-conversion problem, 25¢/min = $15/hr
]
The key to doing a rate problem is recognize that you have a rate problem,
determine which quantities you have and which you want.
Example: A spacecraft is a distance of 1 AU from Earth (AU = astronomical
unit = Earth-Sun distance = 1.5 x 10^11 m) and is approaching at a velocity of 200,000
km/hr (its speed). How long until the spacecraft reaches Earth?
Solution: Amount = distance = 1.5 x 10^11 m
Rate = velocity = 200,000 km/hr
Time = ?
Time = Amount/Rate = 1.5 x 10^11 m/200 000 km/hr (1 km/1000 m) = 750 hr = 31 1/4
days
Do these practice problems:
10. You drive 60 miles per hour for 90 minutes, how many miles do you travel?
11. A black hole swallows up 0.01 solar masses of matter every year, how
many years until it has swallowed 1 solar mass? Note: You don't have to
know what a solar mass is in terms of other units, the solar mass units
will cancel out.
Answers
Practice Problem
Answers
1. 4.48 x 10^-2 = .0448
2. 2 x 10^5
= 200 000 = 2 05 on calculator.
3. 1.67 x 10^2 = 167
How to do #3 on calculator:
6 EE 4 +/- / ( 1 EE 3 x 3 6 EE 1 0 +/- ) =
[Calculators vary so you may have to do something different.]
4. x = (4 hr) (20 dollars) / (8 hr) = 10 dollars
5. x = (100 km) (360·) /
(45·) = 800 km
6. 25.64 g = 25.64 g (1 kg/1000 g) = .02564 kg
7. 9.8 m/s^2
= 9.8 m/s^2
(1 ft/.3048 m) = 32.2 ft/s^2
8. 5 x 10^9 in^2 = 5 x 10^9 in^2 (1 ft/12 in) (1 ft/12 in) = 3.47 x 10^7 ft^2
9. 19.3 g/cm^3 = 19.3 g/cm^3 (1 kg/1000 g) (100 cm/m) (100 cm/m) (100 cm/m)
= 19 300 kg/m^3
10. Distance = Rate x Time = 60 miles/hr (90 min) (1 hr/60 min) = 90 miles
[A very common error is to calculate the result this way:
60 mi/hr becomes 60 mi/(1.5) = 40 mi
The student has divided by 1.5 when multiplying was called for.
The student here has just replaced the hour unit with the number of hours,
1.5.
This, of course, is not how one solves a distance, velocity, time problem
and the answer is wrong.
When you know you have a distance, velocity, time problem and that distance=rate
x time, it is clear to multiply by the time, not divide. The units also
provide a clue, you need multiply 60 mi/hr by some number of hours so that
the hours units will cancel out and you will be left with the number of
miles. ]
11. Rate = .01 sm/yr Time = ? Total amount = 1 sm
Time = Amt/Rate = (1 sm) / (.01 sm/yr) = 100 years