Well, that is a very good question indeed. Let
me begin by asking another question. If I were to tell you that a
dog is 12 long, would you really know how long that dog was?
Now, if I were to tell you that the dog was 12
inches long, you would have a much better idea of the size of the
dog. That, in essence, is what units give us. They give us a
common standard to which we can compare other things. Since we
all know how long one inch is, we can use that knowledge to see
how long a 12 inch dog is.
Most of us know English units fairly well
(assuming you are in the USA), but they are not the most ideal
units to work in because there are some weird units such as
bushels that just don't make any sense to most folks.
This is why we are going to use the metric
system. Or, in your best Inspector Clusso impersonation, Systeme
Internationale. There are some obvious accents missing, but I
didn't take no French when I was in high school. You might be
surprised to learn that the US is officially a metric country
now, but I guess old habits die hard.
So, the units we will use from now on
will all be in S.I. units, which is short for Systeme
Internationale.
- Basic Units
- Length
The unit that we measure length in is meters. In
most cases, we will use "m" for short.
- Mass
The unit that we measure mass in is kilograms.
"kg" for short.
- Time
The unit that we measure time in is seconds.
"sec" for short.
You might be surprised to know that, with
these three units, we can derive the majority of the
other units that we need.
For instance, if you want the units for
area, it is just "m*m", where the "*"
stands for multiplying. This should be obvious because,
for example, the area of a rectangle is given by the
formula, length*width. Since length and width would just
be measured in meters, the result is that area has units
of "m*m".
Likewise, since the volume of a box can
be determined by the formula, length*width*height, it
should not come as a surprise that the units for volume
is "m*m*m".
As you have probably guessed by now, we can
just derive the units of a new quantity by looking at the
formula or how it is defined. Let us do that for several
of the things that we have already talked about. Namely,
let us do this for speed, acceleration, and force.
- Derived Units
- Speed
Recall
the definition of average speed. Average
speed is defined as distance over time. Since
average speed is just distance over time, the
units for average speed would be
"m/sec". "m/sec" is also the
unit for instantaneous speed.
Remember, we are now using S.I. units. An old
familiar unit of speed would be "miles per
hour" or, alternatively, "miles over
hour". If you will compare this with
"meters over second" or
"m/sec" for short, you will see that
they are in the same form, just different units.
Both are distance over time.
Thought Question: What is the
unit of velocity?
To answer this question, think of the
difference between speed and velocity. They
differ only in that we need to specify a
direction when we are talking about velocity.
Therefore, both speed and velocity have the same
units, namely "m/sec".
- Acceleration
Recall
the definition of acceleration.
Since acceleration is defined as the change in
velocity over time, we can just write the unit of
acceleration as the unit of velocity over the
unit of time.
Since the unit of speed, "m/sec", is
the same as the unit of velocity, the unit for
acceleration would be "(m/sec)/sec".
Alternatively, we can write it as
"m/(sec*sec)". Either way is fine.
- Force
To figure out the units of force, we will need to
recall the formula for force given by Newton's 2nd Law.
The formula is
F = ma.
Therefore, the units of force are mass times
acceleration. We already know the units for mass,
and we just figured out the units of
acceleration. Therefore, the unit for force is
"(kg)*[m/(sec*sec)]". Alternatively, we
may write the unit for force as
"(kg*m)/(sec*sec)".
At this point, you might recall that we had an
easier unit for force, namely the
"Newton". If you recalled this, you are
correct. However, there are no tricks here.
"Newton" is just short for
"(kg*m)/(sec*sec)". That's why we used
the "Newton" because it was a lot easier
to just write N instead of (kg*m)/(sec*sec). When
we write "N" for the unit of
force, we are implicitly writing
(kg*m)/(sec*sec).
As an aside, the English unit for force is pounds.
Both pounds and Newtons are units of force. The only difference is that pounds
is the unit of force in the English systems of units while Newtons is the
unit of force in the S.I. system of units. Even though they are both units of
force, they are not equal to one another. In fact, 1 Newton = 0.2248 pounds.
- Converting Units
Now, if everything in the world was easy, then we would only have to deal with one set of units. Alas, such is not the state of the world, and on occasion, we will have to convert units from one system to another.
One example of such a conversion is finding out the metric equivalent of 12 feet. In order to do this, we need to first know the correct unit conversion factor. Well, it turns out that 1 feet is equal to 0.3048 meters.
Conversion factor: 1 feet = 0.3048 meters
Converting units is not a hard thing to do. In fact, it really just involves multiplying and dividing. The best way to learn is to do an example, so here it goes.
-
Example 1: How many meters does 12 feet equal?
The procedure we follow here will be the general procedure used in converting units.
In general, to convert units, we need to multiply the quantity we want to convert (12 feet in this case) by its conversion factor. The conversion factor basically tells us how to convert one unit into another.
Referring to the picture above, you will see the conversion factor is basically a fraction we multiply to the quantity we want to convert. In this case, we know 1 feet = 0.3048 meters. As a result, we place the 0.3048 meters on the top portion of the fraction, and we place the 1 feet in the bottom portion of the fraction. Technically, the top portion of the fraction is called the numerator and the bottom portion is called the denominator.
So, how did we know to place the 0.3048 meters in the top and the 1 feet in the bottom portion of the fraction? Well, in a sense, we had to do this because we needed to cancel the "feet" in the "12 feet". To cancel the "feet" in the "12 feet", we had to divide it by the "1 feet" in the conversion factor. Since we want to convert "feet" into "meters", we needed to cancel the "feet" in the "12 feet" and be left with "meters". This is precisely what we did above. In a sense, we are multiplying and dividing the units just like we do with numbers.
I have done so explicitly in the picture above. You will see that the "feet" cancel because they are divided by each other thus canceling each other. The only unit we are left with is "meters" which is precisely what we want because we wanted to convert the 12 feet into meters. In the picture below, I have just multiplied and divided the units exclusively without the numbers in order to make it a little more clear. You should notice that we are left with the desired unit of "meters" after all the multiplying and dividing.
Let me describe what we just did in words. First we take the "feet" and multiply it by "meters" and then divide it by "feet". As a result, the "feet" divided by "feet" cancel each other out, and we are left with just "meters".
Now that we have done it correctly, let's do it the wrong way and see the result we get. Instead of putting the 0.3048 meters on top and the 1 feet on the bottom, let's pretend we made a mistake and put the 1 feet on top and the 0.3048 meters on the bottom. Remember to also multiply and divide the units as well as the numbers.
Well, you see that putting the 1 feet on top and the 0.3048 meters on the bottom just doesn't work. None of the units cancel. In fact, we are left with a weird unit (feet*feet/meters) which wasn't the one we were after.
Okay, so we've done one example. Let's do another one just to make sure we have this down pat.
-
Example 2: How many feet does 3 meters equal?
Remember, to convert this, all we have to do is multiply the "3 meters" by the appropriate conversion factor. The tricky part is what to put on top and what to put on the bottom in the conversion factor. Try doing this on your own first. One hint you might need is that the conversion factor of 1 feet = 0.3048 meters works for both converting feet to meters and meters to feet. This should be fairly obvious.
The answer you should get is that 3 meters is equal to 9.8425 feet. If you didn't get this answer, try it a couple of times and redo example 1 above before looking at the hint below.
If you will notice in the hint above, we used basically the same conversion factor we used in example 1 above with the exception that "1 feet" is now on top and "0.3048 meters" is now on the bottom.
If you are still having some trouble with this, try redoing example 1 above and apply the same line of reasoning in this example. In addition, you might want to multiply the "3 meters" by the incorrect conversion factor where "0.3048 meters" is on top and "1 feet" is on the bottom. If you do this, you will realize that this is the incorrect conversion factor because it does not leave you with the desired unit of "feet" at the end. In fact, it will leave you with the undesired unit of "meters*meters/feet".
If you fully understand what you just did, then you are well on your way to understanding how to convert units. Next, let's try converting something a little more difficult.
-
Example 3: What is 55 mph in terms of meters/sec?
First, let's write 55 mph as 55 miles/hour. In this form, we see that, in order to convert it to "meters/sec", we need to convert the "miles" on top to "meters", and we need to convert the "hour" on bottom to "seconds". So, it turns out that it really isn't too difficult. We are just converting two units instead of one.
To do this example, you will need to know the following two things.
1 mile = 1609.344 meters and 1 hour = 3600 seconds
You probably already knew the second fact above because there are 60 minutes in an hour and 60 seconds in every minute. Now you know everything you need to know. Try doing the problem on your own first. All you have to do is be a little careful and remember to use two conversion factors. The answer you should get is that 55 miles/hour is equal to 24.5872 meters/sec.
As stated above, we needed two conversion factors, which we multiplied to the 55 miles/hour. The first conversion factor converted the "miles" on top to "meters". The second conversion factor changed the "hour" on the bottom to "seconds".
The converting "miles" to "meters" should have been fairly easy because it was very similar to what we did in example 1 and example 2 above. The only thing tricky might have been the second conversion factor which converted the "hour" on the bottom of 55 miles/hour to "seconds" on the bottom in the final answer. This shouldn't have posed too much of a problem if you were already comfortable with converting units. The main thing to remember in the second conversion factor is that we wanted to end up with "seconds" on the bottom and not on the top.
This last example might have been confusing. If you are still having trouble with it, take your time with it. If you are still getting the incorrect answer, make sure to take a look at your units. If the units you end up with are not "meters/sec" and you have some weird leftover units, then what you did was incorrect. The whole point of converting units is to end up with the correct units. If you set up the conversion factors so that the units you end up with are correct, the number answer you get from that should also be correct.
-
Example 4: Convert 10 meter/sec to miles/hour
If you were able to do example 3 above, this shouldn't be any trouble at all. It's mostly just for practice. If you still aren't comfortable with converting units, make sure you understand example 3 above before proceeding with this problem.
Once again, try to do this on your own first before looking at the hint. The answer is 22.369 miles/hour.
So, why this digression into units? Well, we are going to have
to use them in the next section where, as promised, I will reveal
the very useful formula that will make the problems in the Constant Acceleration
section above a lot easier.
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