Physics I For Dummies. Steven Holzner
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Suppose you want to convert 3 days to hours. You can just multiply your time by the preceding fraction. Doing so doesn’t change the value of the time because you’re multiplying by 1. You can see that the unit of days cancels out, leaving you with a number of hours:
To convert the other way — hours into days, in this example — you simply use the same original relation, 24 hours = 1 day, but this time divide both sides by 24 hours to get
Then multiply by this fraction to cancel the units from the bottom, which leaves you with the units on the top.
Consider the following problem. Passing the state line, you note that you’ve gone 4,680 miles in exactly three days. Very impressive. If you went at a constant speed, how fast were you going? Speed is just as you may expect — distance divided by time. So you calculate your speed as follows:
Your answer, however, isn’t exactly in a standard unit of measure. You have a result in miles per day, which you write as miles/day. To calculate miles per hour, you need a conversion factor that knocks days out of the denominator and leaves hours in its place, so you multiply by days/hour and cancel out days:
Your conversion factor is days/hour. When you multiply by the conversion factor, your work looks like this:
Note that because there are 24 hours in a day, the conversion factor equals exactly 1, as all conversion factors must. So when you multiply 1,560 miles/day by this conversion factor, you’re not changing anything — all you’re doing is multiplying by 1.
When you cancel out days and multiply across the fractions, you get the answer you’ve been searching for:
LOOKING AT THE UNITS WHEN NUMBERS MAKE YOUR HEAD SPIN
Want an inside trick that teachers and instructors often use to solve physics problems? Pay attention to the units you’re working with. Considering the thousands of one-on-one problem-solving sessions with students in which we worked on homework problems, this is a trick that instructors use all the time.
As a simple example, say you’re given a distance and a time, and you have to find a speed. You can cut through the wording of the problem immediately because you know that distance (for example, meters) divided by time (for example, seconds) gives you speed (meters/second). Multiplication and division are reflected in the units. So, for example, because a rate like speed is given as a distance divided by a time, the units (in MKS) are meters/second. As another example, a quantity called momentum is given by velocity (meters/second) multiplied by mass (kilograms); it has units of kg · m/s.
As the problems get more complex, however, more items are involved — say, for example, a mass, a distance, a time, and so on. You find yourself glancing over the words of a problem to pick out the numeric values and their units. Have to find an amount of energy? Energy is mass times distance squared over time squared, so if you can identify these items in the question, you know how they’re going to fit into the solution and you won’t get lost in the numbers.
The upshot is that units are your friends. They give you an easy way to make sure you’re headed toward the answer you want. So when you feel too wrapped up in the numbers, check the units to make sure you’re on the right path. But remember: You still need to make sure you’re using the right equations!
So your average speed is 65 miles per hour, which is pretty fast considering that this problem assumes you’ve been driving continuously for three days.
You don’t have to use a conversion factor; if you instinctively know that you need to divide by 24 to convert from miles per day to miles per hour, so much the better. But if you’re ever in doubt, use a conversion factor and write out the calculations, because taking the long road is far better than making a mistake. Far too many people get everything in a problem right except for this kind of simple conversion.
Eliminating Some Zeros: Using Scientific Notation
Physicists have a way of getting their minds into the darndest places, and those places often involve really big or really small numbers. Physics has a way of dealing with very large and very small numbers; to help reduce clutter and make them easier to digest, it uses scientific notation.
For example, say you’re dealing with the average distance between the sun and Pluto, which is about 5,890,000,000,000 meters. You have a lot of meters on your hands, accompanied by a lot of zeroes. You can write the distance between the sun and Pluto as follows:
The exponent is 12 because you count 12 places between the end of 5,890,000,000,000 (where a decimal would appear in the whole number) and the decimal’s new place after the 5.
Scientific notation also works for very small numbers, such as the one that follows, where the power of ten is negative. You count the number of places, moving left to right, from the decimal point to just after the first nonzero digit (again leaving the result with just one digit in front of the decimal):
Here’s a simple example: How does the number 1,000 look in scientific notation? You’d like to write 1,000 as 1.0 times ten to a power, but what is the power? You’d have to move the decimal point of 1.0 three places to the right to get 1,000, so the power is three:
USING UNIT PREFIXES
Scientists have come up with a handy notation that helps take care of variables that have very large or very small values in their standard