From meters to inches and back again: Converting between units
Physicists use various measurement systems to record numbers from their observations. But what happens when you have to convert between those systems? Physics problems sometimes try to trip you up here, giving you the data you need in mixed units: centimeters for this measurement but meters for that measurement — and maybe even mixing in inches as well. Don’t be fooled. You have to convert everything to the same measurement system before you can proceed. How do you convert in the easiest possible way? You use conversion factors, which we explain in this section.
Using conversion factors
Here’s how it works: For every relation between units — for example, 24 hours = 1 day — you can make a fraction that has the value of 1. If, for example, you divide both sides of the equation 24 hours = 1 day by 1 day, you get
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.
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