Any calculation of time (t) is relatively easy in the cases where either vi or a was zero. In the cases where neither are zero, the math results in a second degree polynomial equation such as:
0 = x2 + 3x - 12
Where this occurs in physics of motion is in the full distance equation:
df = di + (vi)(t) + (1/2)(a)(t2)
Although this does not exactly match the expected form for a quadratic equation, it can easily be rearranged as such:
df = di + (vi)(t) + (1/2)(a)(t2)
0 = di - df + (vi)(t) + (1/2)(a)(t2)
0 = (1/2)(a)(t2) + (vi)(t) + di - df
Keep in mind that di - df will yield a number when the values are plugged in and simplified.
To solve these problems, the steps are the same for any problem in science, BUT the algebra becomes harder.
After you have plugged in the numbers, combine like terms and simplify. Suppose the following:
df = 89
di = 50
vi = 4
a = 6
and you need to find t
df = di + (vi)(t) + (1/2)(a)(t2)
89 = 50 + (4)(t) + 1/2(6)(t2)
0 = -39 + 4t + 3t2
(Put it in normal quadratic form.)
0 = 3t2 + 4t - 39
Now you can either factor or use the quadratic formula to find the values for t:
0 = ( t - 3 )( 3t + 13)
0 = t - 3 AND 0 = 3t + 13
3 = t AND -13/3 = t
Since time cannot be negative within the context of classical physics, only t = 3 is a valid answer.
Using the quadratic equation will yield the same results.
While it is far easier to find t when either a or vi is zero, the math to find t when that is not the case is not beyond the skills of a student taking an introductory physics class.
For another look at this process, check out this video:
Equations numbers are just for conversations… They don't imply any meaning.
#1 :df = di + (vi)(∆t) + (½)(a)(∆t2) This will answer almost all “how far” questions.
Use this if: Only 1 velocity is given You have a distance given (df is rarely 0, but sometimes it is the unknown)
#2 :df = di + (vi)(∆t) This is the simplified version of the previous equation.
Use this if:
There is no acceleration given, but you are finding distacne/displacement.
#3 :∆d = v(ave)∆t
Use this if: You only need to find how much something moved in a period of time, not how far it ends up from a point of reference.
#4 : vf = vi + (a)(∆t) This will answer almost all “how fast” questions.
Use this if: 2 velocities are given You don’t have distance
#5 : ∆v = a∆t
Use this if: You only need to find how much velocity changed over a period of time, not what its final or initial velocity was/is.
#6 : v(ave) = (vi + vf)/2 (See * below)
Use this if: You need to find the average velocity You know BOTH vi and vf (you might have to calculate vf!
#7 : vf = (vi + ∆v)
Use this if: You need to find the final velocity AND You know BOTH vi and how much v changes (e.g. speed increases by 10 m/s)
#8 : df = (di + ∆d)
Use this if: You need to find the final distance, total distance, or final position AND You know BOTH di and how much d changes (e.g. it moves 10 meters)
#9 : F = ma
Use this if: You are given force and mass and need acceleration…
The hints and helps:
df = final distance
di = initial distance
vf = final velocity
vi = initial velocity
v(ave) = average velocity
∆t = elapsed time
∆d = change in displacement
∆v = change in velocity
m = mass
a = acceleartion
F = force
It is vi if…
“…traveling at a rate of…”
“…moving at…” "…has a velocity of…"
If it is “at rest” then vi = 0 and di is probably 0
The words "begin" and "starts" generally go with the initial values.
If it is “moving at a constant” rate or if it “has a constant velocity” then a = 0
It is possible that something not given (but which is not the thing to be found) should have a value of 0...
*Another v(ave) approach:
v(ave) = ((vi + at) + vi)/2
Use this if:
You need to find the average velocity
You know vi and but you have to calculate vf
The 21st Century understanding of atomic theory is very effective in helping explain and predict how substances interact. It is the result of a growing body of knowledge that dates back centuries.
In the middle of the 5th Century BCE, Democritus proposed ideas that were correct in many ways. Democritus believed that all matter consisted of extremely small particles that could not be divided. He called these particles atoms from the greek word transliterated as atomos, which means "uncut" or "indivisible."
He thought there were different types of atoms with specific sets of properties for the different substances in creation. The atoms in liquids, for example, were round and smooth, but the atoms in solids were rough and prickly.
Though Democritus's ideas didn’t catch on, he was surprisingly accurate in his assumptions. Scientist of the 21st Century eventually concluded that many of Democritus’s ideas were correct, and they extended the understanding of atomic theory in stages that led to the current model.
John Dalton
Dalton proposed the theory that all matter is made up of individual particles called atoms, and in it, he identified several things that have lasted into the modern atomic theory:
All elements are composed of atoms.
All atoms of the same element have the same mass, and atoms of different elements have different masses.
Compounds contain atoms of more than one element.
In a particular compound, atoms of different elements always combine in the same ratios.
J. J Thompson
Thompson's experiments provided the first evidence that atoms are made of even smaller particles that have positive and negative charges.
Ernest Rutherford
Rutherford's model extended what was previously understood by identifying that atoms have a central, relatively dense (compared to the entire volume of an atom) nucleus around which electrons move.
Niels Bohr
In Bohr's model, electrons move with constant speed in fixed orbitals around the nucleus.
Electrons must orbit the nucleus in one of several fixed, specific orbits, and each orbit represents a specific energy level.
The first orbital represents the lowest electron energy level, and the other orbitals represent progressively higher and higher energy levels.
Electron Cloud Model
Evidence following Bohr's work led to the understanding that the electrons do not orbit the nucleus like a planet.
While they do exist at specific energy levels and occupy orbitals, their position in the orbital is never 100% certain. They are somewhere in the orbital, but exactly where cannot be known specifically. Each orbital can be, therefore, conceived as an electron cloud.
The progression of Atomic Theory in the modern era took place in various steps and stages. The resulting body of understanding results in a model of atoms that is highly effective and useful in understanding and predicting the behavior of substances.
Throughout science, there inevitably comes a time when it is necessary to use some "known relationship" to figure out what happened or what might happen. In physics and chemistry, especially, this is the case, but sciences like sociology use "known relationships," such as population growth models, to explain or predict certain happenings. Business science looks at factors such as supply and demand or production cost vs sales price as means of maximizing profit. In many, many cases, the "known relationship" is expressed as a mathematical equation. The relationship between the rate something happens and the amount of something produced is very common, so this discussion will use a couple of rate problems as its example.
In most general terms, the amount of something produced is equal to the rate of production times the amount of time production took place. For instance, in a displacement problem, rate is how far something moves in a given time and motion of an object is produced. In a general form a rate problem might look like this:
O = R•t
where O is the output, R is the rate, and t is the time.
So what is this process of solving science problem that will use rate problems as its example?
There are three steps to solving ANY science problem (or word problem of any kind, for that matter).
The following two examples will be used as the steps are discussed:
EXAMPLE 1: A baker's oven will hold only 1 pan of cookies, and each pain has space for 12 cookies. The baking time on cookies (including putting the dough on the pan) is 20 minutes. Therefore, the rate cookies are baked is 12 cookies per 20 minutes. How many cookies can be baked in 80 minutes?
EXAMPLE 2: A car travels at an average rate of 20 MPH for 3 hours. How far does it go.
The first stepis to write down what is asked for. What are you supposed to find?
Example 1:
LOOKING FOR "How many cookies"
Example 2:
LOOKING FOR "How far does it go"
A student familiar with physics motion problems would see Example 2 as a "distance" problem and would use the distance equation variables. In such a case, R would be velocity, v, t would be time, t, and LOOKING FOR would be distance, d (or in some cases, s).
The second step is to write down what's given and find an equation that relates to the question.
Example 1:
Given:
Rate of baking (R) = 12 cookies/20minutes
time (t) = 80 minutes
Intuition and general math instruction will lead you to a formula where…
Number of Cookies = R • t
Example 2:
Rate of motion (v) = 20 MPH
time (t) = 3 hours
Intuition or a quick Google search will reveal a formula (the distance formula) that related velocity and time of motion. d = vt
From time to time, it is necessary to rearrange an equation so that it yields the answer you are trying to find. For example, if you trying to find time in Example 2 above, you will need to isolate the t variable. MANY TIMES it is easier to do the algebra BEFORE you plug in the numbers and units!
d = vt (To find t divide both sides by v.)
d/v = vt/v (The v on the right side cancels.)
d/v = t
Once you have an equation in the form you need, you are ready for the next and final step.
The third step is to plug in what was given and solve. Assuming the second step was completed, the following solutions would emerge:
Example 1:
Cookies = R•t
Cookies = 12 Cookie/20 min • 80 min
Cookies = 960 Cookie•min / 20 min
Cookies = 48 Cookie
Example 2:
d = vt d = 20 mi/hr * 3hr d = 60 (mi•hr)/hr d = 60 mi
For additional examples and a video explanation, check this out:
Scan for Video
Scan for Video
Example 3 (All together)
A car ball rolls at a constant rate of 20 m/s. How far will it roll in 8 seconds?
Using df as final distance rolled, v as the velocity, and t as time…
Find df where
v = 20 m/s t = 8 s
df = vt df = (20 m/s) (8 s) df = 160 m
SUMMARY
There are three steps to solving problems.
1: Write down what is asked for.
2: Write down what is given and find a relevant equation.
2b: Rearrange the equation so that it yields the answer you want.
3. Plug in and solve.
The same thing expressed as five steps:
1 Write down what is being asked for.
2. Write down what is given.
3. Identify a relevant equation.
4. Plug in.
5. Solve.
Objects in the physical world can be described according to various physical properties, such as length, volume, mass (or weight). Since the beginning of humanity, people have come up with ways to measure things in a standard way.
Standards and Units
Measurements were usually created to compare to some reference object or other agreed-upon standard. For instance, if trading sea shells, the standard would be… a seashell. One sea shell equaled one seashell.
Okay… hang in there… keep going…
So, along the way, magic occurred (not really) and people began to equate the word "one" with "unit." So, if you said, "Give me eight units of seashells," since seashells were measured in… themselves… you would get eight seashells.
But suppose a guy sold sand. Selling grains of sand would be… dumb. Suppose (sticking to the beach motif) he had a coconut hollowed out. His thing was, one scoop of sand for one seashell…
So for sand, the unit would be scoops.
A buyer would say, "Give me eight scoops of sand," and would pay eight seashells for it.
It must be time for a definition!
Unit: a quantity chosen as a standard in terms of which other quantities may be expressed. (Oxford dictionary.
Not very helpful, but in the right direction!
UNIT DEFINITION A unit is any standard used for comparison in measurements.
Going back to strictly intuition, most people already know what units are. "Comparison" is a good word to hang onto, so do that.
Now think about things you already know. Gallons of gasoline. Pounds of lunch meat. Miles to the next town.
Using a standard unit like miles allows us to make reasonable comparisons. A mile is a mile is a mile. So if it is 200 miles to Townville and 400 miles to Villaton, since a mile is a mile is a mile, it must be twice as far to Villaton.
If I have 4 gallons of paint and you have 2, I have twice as much paint, since a gallon is a gallon is a gallon.
So, think about things around you. What are the standard units for them.
Distances between towns?
Shampoo?
Foundation?
Firewood?
Soda (pop)?
Chances are, for at least one of those, you thought of different standards. For example, you might find 20 oz bottles of soda as well as 2 liter bottles!
There are times when converting from one standard unit to the other is desirable.
Conversion and Equivalency
The process of converting from one standard unit to another boils down to finding how many of one thing is equal to one thing of the other.
Good news, #1: The math on this is easy.
Good news, #2: You can usually look it up on Google.
Bad news, #1: An explanation follows anyway.
To start with a definition of conversion…
Conversion: The process of finding out how many units from one standard are equal to how many units from another standard.
Suppose you discover that, in a fantasy novel, the people in one town sell milk by the Nallog and in another by the Ecnuo. Some guy has a barrel on which are etched lines for both Nallogs and Ecnuos, and another dude sees that the milk comes to the 2 Nallog mark and also the 256 Ecnou mark.
In the (silly) example, you can see that 2 Nallogs = 256 Ecnous. That is, the two quantities are equivalent. Finding out how many Ecnous are in a Nallog is easy! Divide the bigger number by the smaller. Bam!
256 ÷ 2 = 128
So 1 Nallog (the bigger unit) is equivalent to 128 Ecnou (the smaller unit).
In the (silly) example above, the conversion factor is 2.
Conversion factor:A number that, when multiplied will convert one unit to another.
Oxford Dictionary says it like this: an arithmetical multiplier for converting a quantity expressed in one set of units into an equivalent expressed in another.
To find (or derive) a conversion factor for anything, all that is needed is to know how much of something is present in the two different units. Then, to find the conversion factor, divide the larger by the smaller of the quantities. The quotient (number after you push the equal button) will be how many of the smaller thing (the bigger number) are in the bigger thing (smaller number).
No, that's not confusing, is it?
EXAMPLE
A container has 591.5 milliliters of shampoo in it. The label also says it has 10 clarkens (which are cleverly abbreviated as "Clar"). Find the conversion factor for milliliters and clarkens.
591.5 ml
_______ = 59.15 ml/Clar
10 Clar
Thus, 1 Clar is equal to 59.15 ml.
BAM! Easy math!
Conclusion
It is always best to just measure things in the units you need. But sometimes, that's impossible. Sometimes your most accurate measuring device provides units that you have to convert.
A solid understanding of equivalency and conversion prepares students to face the demands of science and engineering.
About that math…
Use the following link to see an example of how to do the math for unit conversions…
Where are we going with this? This page will assist in understanding the relationship between units and the quantities they measure.
When looking at the world around, people pretty much automatically attend to physical quantities, pretty much without thinking about them. The youngest child intuitively can judge differences with some skill.
By the time the child reaches school age, the following conversation would seem drastically out of place:
Teacher: "Mary, how far can you run?"
Mary: "I can run six pounds! Very heavy!"
Almost all children would know that far and heavy describe different physical quantities.
As most readers of this would know, one of the jobs of science is to complicate things. No… not that…
to clarify things by assigning specific words with specific meanings to specific ideas, objects and concepts in order to make discussion precise and accurate (which are both words surrounded by confusion, ironically).
Therefore, with regard to physical quantities, a certain set of words are used to describe specific things.
Included in those sets of words are two sides to every quantity.
One side is the concept of what is being described. Mary, in the dialog above, gave an answer about one quantity with words used for a different.
The other side of the concept is, for every quantity, specific ways that it is measured (which are called units). Pounds do not go with a "how far" question!
First what are the things being described?
Types of Physical Quantities
The following list is certainly not all-inclusive. It includes a few very common and familiar quantities that are measured.
Distance: Distance is a measure of… okay, distance is so common it actually has different meanings, so in science, there are different words that are used so in order to be more specific.
Distance (Most General): The measure of how far two points are from each other, as in, "The tip of the antenna was 12 meters from the the surface of the window."
Distance (A specific case) ThinkDistance Traveled:The length of a path that a moving body takes, as in, "The dog ran from tree to tree through the park until it was at the tree next to the one where it started, covering a distance of 250 yards." Compare with Displacement, next!
Displacement: The measure in a straight line between where a moving body begins and ends, as in, "The dog ran from tree to tree through the park until it was at the tree next to the one where it started, resulting in a displacement of only 4 yards."
Length, Width, Height: Pretty much what you expect, on this! The measure of how far specific points of an object are from other points on the object.
Other things that would be included in the distance concept are circumference, radius, perimeter, range, altitude, depth… You can probably think of others. But will you?
In general, distance is the measure of how far apart two things are.
Volume: (Not talking about sound, which borrowed from this concept for its own purposes!) This also is pretty much what you expect!
Volume is the space in three dimensions that an object or substance takes up (or holds). If a tank holds 25 gallons of gasoline, then the volume of the gasoline that filled the tank would be (duh) 25 gallons. A bottle of soda holds 20 ounces. A different bottle holds 2 liters.
Mass: Mass is a physical quantity that is the result of how many protons, electrons, and neutrons are all in a given space. It is actually not directly observable, though there are devices (scales and balances) that combine with gravity (or other acceleration) and other laws of physics so that it can be measured. Mass is a measure of the amount of matter an object contains.
If two things with the same volume have different masses, one would feel heavier than the other.
Good news: There are scales that measure mass, so you can just plop things down and get a number!
Weight: Weight is a measure of mass under a particular condition. Many people have heard things like, "Well, on the moon, I'd only weigh 12 pounds."
Weight is a basic concept that people are very familiar with. Weight is the degree of heaviness something has.
Time: This is something people very intuitively understand, but which is actually a very abstract concept. Stop reading and write down a definition of time. Not how time is measured! Try a definition of time that does not use seconds, minutes, hours, days, etc. and see what you come up with!
According to the Oxford dictionary, time is the indefinite continued progress of existence and events in the past, present, and future regarded as a whole.
Time is a very basic concept in science, and fortunately, our intuitive understanding is enough for us to use it.
How do you measure these quantities?
The list below will connect the quantities above with the SOME OF THE units used to measure them. Common units in science are in bold and "normal" abbreviations for the common units are in parenthesis..
Mass: slugs, grams (g or gr), kilograms (kg) Time: hours (hr), minutes (min), seconds (s)
A few more…
Some units are derived or combined from the basic units but are so common they are worth noting here.
Force: Newtons (n or N), Pounds
Weight (which is a force) is also measured in pounds, newtons
Speed or Velocity: MPH, m/s, cm/s, km/hr Speed or Velocity is a distance unit divided by a time unit.
Acceleration: (m/s)/s, MPH/s, (km/s)/s, cm/s2m/s2
Acceleration is a velocity unit divided by a time unit.
At a glance…
System
Mass
Weight
Distance
Time
Acceleration Due to Gravity
English
Slug
Pound lb
Feet
ft
Seconds
s
32 f/s/s
Metric
Kilogram kg
Newton
N
Meters
m
Seconds
s
9.8 m/s/s
And also…
Temperature: Temperature is, within the kinetic theory of matter model, defined as the average kinetic energy of the molecules in a system or substance. How much thermal energy is present? How hot is it? The absence of thermal energy is described as being cold. Temperature is measured in degrees. In science, officially in Kelvin degrees (°K), but often in Celsius degrees (°C).
SI Units
With so many units, things could get confusing. So… Some smart people made a decision that, a a general best-practice science would be built on seven base units. They also gave it a fancy name:
The International System of Units
Then… they decided they would abbreviate it… from French
Système international (d'unités)
The seven base SI units and some corresponding constants allow meaningful and uniform dialog about all matters of science.
