U Can: Physics I For Dummies. Steven Holzner

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:

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. We’ve had thousands of one-on-one problem-solving sessions with students in which we worked on homework problems, and we can tell you that 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!

      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:

      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. We’ve seen far too many people get everything in a problem right except for this kind of simple conversion.

      Here are some handy conversions that you can come back to as needed:

      ✔ Length:

      ● 1 m = 100 cm

      ● 1 km = 1,000 m

      ● 1 in (inch) = 2.54 cm

      ● 1 m = 39.37 in

      ● 1 mile = 5,280 ft = 1.609 km

      ● 1 Å (angstrom) = 10^– 10 m

      ✔ Mass:

      ● 1 kg = 1,000 g

      ● 1 slug = 14.59 kg

      ● 1 u (atomic mass unit) = 1.6605 × 10^– 27 kg

      ✔ Force:

      ● 1 lb (pound) = 4.448 N

      ● 1 N = 10⁵ dynes

      ● 1 N = 0.2248 lb

      ✔ Energy:

      ● 1 J = 10⁷ ergs

      ● 1 J = 0.7376 ft-lb

      ● 1 BTU (British thermal unit) = 1,055 J

      ● 1 kWh (kilowatt hour) = 3.600 × 10⁶ J

      ● 1 eV (electron volt) = 1.602 × 10^– 19 J

      ✔ Power:

      ● 1 hp (horsepower) = 550 ft-lb/s

      ● 1 W (watt) = 0.7376 ft-lb/s

       Examples

      Q. A ball drops 5 meters. How many centimeters did it drop?

      A. The correct answer is 500 centimeters. To perform the conversion, you do the following calculation:

      Note that 100 centimeters divided by 1 meter equals 1 because there are 100 centimeters in a meter. In the calculation, the units you don’t want – meters – cancel out.

      Q. Convert 10 inches into meters.

      A. The correct answer is 0.254 meters.

      1. You know that 1 inch = 2.54 centimeters, so start with that conversion factor and convert 10 inches into centimeters:

      2. Convert 25.4 centimeters into meters by using a second conversion factor:

      Q. An SUV is traveling 2.78 × 10^– 2 kilometers per second. What’s that in kilometers per hour?

      A. The correct answer is 100 kilometers per hour.

      1. You know that there are 60 minutes in an hour, so start by converting from kilometers per second to kilometers per minute:

      2. Because there are 60 minutes in an hour, convert this to kilometers per hour by using a second conversion factor:

       Practice Questions

      1. How many centimeters are in 2.35 meters?

      2. How many inches are in 2.0 meters?

      3. Given that there are 2.54 centimeters in 1 inch, how many centimeters are there in 1 yard?

      4. How many inches are in an angstrom, given that 1 angstrom (Å) = 10^– 8 centimeters?

      5. How many hours are in 1 year?

      6. The three-toed sloth can move at a speed of about 10 feet per minute. How fast is that in kilometers per hour? (Round to the nearest hundredth of a kilometer per hour.)

       Practice Answers

      1. 235 cm. Convert 2.35 meters into centimeters:

      2. 79 in. Convert 2.0 meters into inches:

      3. 91.4 cm. One yard is 3 feet, so convert that to inches:

      Use a second conversion factor to convert that into centimeters:

      4. 4.0