About 500 years ago it began to be recognized that the universe behaves mathematically. This was a surprising result and still is when you think about it. Nonetheless, it is true: if you are to work with the simplest forms of nature, you must use the language of mathematics. This section is meant to review for you the concepts of algebra and introduce you to some of the vocabulary and tools of physics.

1. A car travels 486 miles in 8.7 hours. What is its average speed?

average speed = 486 mi/8.7 hr average speed = 55.9 mi/hr

2. A snail slides 13 cm in 28 seconds. What is its average speed?

average speed = 13 cm/28 sec average speed = .464 cm/sec

3. An electron travels an astonishing 1.63 meters in 5.7 x l0^-8 seconds. How fast is it moving?

speed = 1.63 m/5.7 * 10-8 sec speed = 2.86 * 109 m/sec

4. Notebook paper measures 8.5 inches by 11 inches. What is its area? (That is, the area of one side of the paper.)

Area = Length * Width A = 93.5 in2

5. A block of iron measures 3.6 cm by 8.4 cm by 4.4 cm. What is its volume?

Volume = l * w * h V = (3.6 cm)(8.4 cm)(4.4 cm) V = 133 cm3

Pay particular attention in these problems to the units. In problem 1 you divide miles by hours to get miles/hr (read this as "miles per hour). In problem 2 you divided centimeters by seconds to get cm/sec. In problem 4 you multiplied inches by inches to get inches² or "square inches". Finally, in problem 5, by multiplying cm by cm by cm, you got cm³ or "cubic centimeters". Keep units in mind as you complete the next set of problems.

6. A car travels at 51.4 mi/hr for 1.6 hr. How far does it get in this time?

Distance (s) = speed * time s = (51.4 mi/hr)(1.6 hr) s = 82.2 mi

7. A piece of wire has a mass of 6.4 gm/cm (grams per centimeter). What is the mass of 13.6 cm of this wire?

mass (m) = (6.4 gm/cm)(13.6 cm) m = 87.0 gm

8. Corn valley has a population of 40,200 and an area of 6.84 mi². What is the population density?

population density = people/mi2 population density = 40,200 people/6.84 mi2 population density = 5.88 * 103 people/mi2

9. Lead has a density of 11.3 gm/cm³. What is the mass of 325 cm³ of lead?

Each cubic centimeter has a mass of 11.3 gm, so 325 cm3 has a mass: m = (11.3 gm/cm3)(325 cm3) m = 3.67 * 103 gm

Don't forget to be aware of units. They behave just as fractions did in basic math. Recall how you learned to cancel: (3/5) * (5/8) = 3/8 (5/16)/(9/16) = 5/9 9 * (5/9) = 5 This same approach may be used successfully on the units of quantities: (2 ft/sec) * (60 sec/min) = 120 ft/min (30 gm/cm2)/(1000 gm/min) = 0.03 min/cm2 6 sec * (4 ft/sec) = 24 ft Study solutions 6 through 9 until the manipulation of units in these problems comes easily to you. It's hard to overestimate the importance of being able to work with units. Here are yet more:

10. Anita Solution works steadily at making a rug. She completes 15 square inches in 37.2 minutes. What is her average rate of work?

average rate of work = 15 in2/37.2 min average rate of work = .403 in2/min

11. C. D. Twain puts on his favorite album, which rotates at 33.3 revolutions/minute. How many times does the disk revolve for a song that lasts 4 minutes 22 seconds?

First express the time in minutes: 4 min 22 sec = 4 + 22/60 = 4.37 min revolutions = (33.3 rev/min)(4.37 min) revolutions = 146

12. Atmospheric pressure is 14.7 pounds per square inch. What is the total force on a table top 27 inches wide by 43 inches long?

Force = (14.7 lb/in2)(27 in)(43 in) Force = 1.71 * 104 lb

Be aware that you do not really need to know what the problem is all about. If you know what units you are supposed to end up with, you can solve the problem by making sure they cancel to give the correct final unit.

13. A block of wood measures 3.5 cm by 8.2 cm by 6.7 cm. Its mass is 131 grams. What is its density?

To find density we must determine how many grams are in 1 cubic centimeter. Thus we take the total mass and divide by the number of cm3. density = 131 gm/[(3.5 cm)(8.2 cm)(6.7 cm)] density = .681 gm/cm3

14. The density of iron is 7.6 gm/cm³. What is the mass of an iron rod 0.44 cm in radius and 25 cm long?

Volume of cylinder = area of base * height Volume of cylinder = (3.14)r2 Mass (m) = density * volume m = (7.6 gm/cm3)[(3.14)(.442 cm2)(25 cm) m = 116 gm

Check your units again. If these problems seem a bit too easy, bear with me a thorough familiarity with these concepts is too important to pass by quickly.

15. A bicyclist travels at 37 ft/sec. How long does it take him to travel 1.0 feet?

We know ft/sec, but we wish to know sec/ft; that is, the number of seconds required to travel 1 foot: time (t) = 1/(37 ft/sec) t = 1 sec/37 ft t = .0270 sec/ft

16. A sound wave vibrates at 13,540 cycles/sec. How long does it take for one vibration?

We know cycles per second, but we wish to know seconds per cycle: t = 1/(13,540 cycle/sec) t = 7.39 * 10-5 sec/cycle

17. Water pressure on a surface is 128 pounds/in². What surface area will feel the force of one pound?

As before: surface area = 1/(128 lb/in2) surface area = .00781 in2/lb

18. A city has a population density of 18,757 people/mi². How much area does this give each person?

Again: Area = 1/(18,757 people/mi2) Area = 5.33 * 10-5 mi2/person This is a square 39 ft on a side!

The problems you've just completed may look a bit trivial, but there's an important lesson. When the number is inverted, the units that go with the number are inverted as well. This will be useful in the following problems.

19. Rain drops fall on a tile surface at a density of 4,675 drops/ft². There are 16 tiles on each square foot of floor space. How many drops fall on each tile?

We wish to end up with units of drops/tile. Thus ft2 must cancel out: drops/tile = (4675 drops/ft2)/(16 tiles/ft2) drops/tile = 292

20. A stretch of desert highway outside Barstow, California still retains Burma-Shave signs from the early 1960's. There are 18 signs/mile, and a car speeds by them at 55 mi/hr.
a) At what rate will the driver see the signs? (That is, how many signs/hour will pass by him?
b) What time interval will pass between successive signs?

a) Miles will have to cancel out: rate of signs = (18 sign/mi)(55 mi/hr) rate of signs = 990 sign/hr note: this does not mean there are 990 signs on the road. Seeing 99 signs in 1/10 hr would do it.

b) interval = 1/(990 sign/hr) interval = 1.01 * 10-3 hr/sign note: this is 3.6 sec between signs

21. One quart of Slopiton paint covers 450 ft². Doris Open wishes to paint baseballs, each of which has an area of 0.12 ft². How many quarts of paint will be required to complete 3,486 baseballs?

We want an answer with quarts for a unit. Baseballs are ft2 will have to cancel out and a quart will need to end up in the numerator. quarts of paint = [(.12 ft2/baseball)(3486 baseball)]/(450 ft2/qt) quarts of paint = .93

By now you should be aware of an important fact: Units can tell you how to solve a problem. If, on a test, you find you can't get your mind off the weekend's date, a little reasoning with units is a great substitute for understanding. This is most easily seen in the conversion of units from one system to another. You will need the following English and metric conversions: 1 inch = 2.54 cm 1 foot = 12 inches 5280 feet = 1 mile 100 centimeters = 1 meter 1 kilometer = 1000 meters 1.057 quart = 1 liter 1 liter = 1000 cm3 1 pound = 4.45 newtons Take the relationship that 1 inch = 2.54 cm. It follows, then, that: 1 in/2.54 cm = 1 2.54 cm/1 in = 1 Since we can multiply or divide any quantity by 1 without changing its value, the following is mathematically permissible: (26.8 cm)(1 in/2.54 cm) = 10.6 in We have shown that 26.8 cm is equivalent to 10.6 inches. Now try to convert 16.0 inches to centimeters. (16.0 in)((2.54 cm/1 in) = 40.6 cm

22. convert 18,722 ft into miles.

(18, 722 ft)(1 mi/5280 ft) = 3.55 mi

23. Convert 3.6 meters into centimeters.

(3.6 m)(100 cm/1 m) = 360 cm

24. Convert 8.6 x l0^-4 cm into inches.

(8.6 * 10-4 cm)(1 in/2.54 cm) = 3.39 * 10-4 in

25. Convert 3.62 quarts (qt) into liters (l).

(8.6 * 10-4 cm)( 1 l/1.057 qt) = 3.42 l

26. Convert 3.86 x 10³ cm³ into quarts.

(3.86 * 103 cm3)[(1.057 qt)/(103 cm3)] = 4.08 qt

In the next problems string together several factors to complete the conversions.

27. Convert 877 cm into miles.

convert cm-->inches-->feet-->miles (877 cm)(1 mi/2.54 cm)(1 ft/12 in)(1 mi/5280 ft) = 5.45 * 10-3 mi

28. Convert 2.644 days into minutes.

convert days-->hours-->minutes (2.644 days)(24 hr/1 day)(60 min/1 hr) = 3.81 * 103

29. Convert 15 in² to cm². (Remember, when you square a unit, you must also square the conversion factor.)

(15 in2)(2.54 cm/1 in)2 = (15)(6.45 cm2) = 96.8 cm2

30. Convert 3.7 x l0-3 ft³ into in³.

(3.7 * 10-3 ft3)(12 in/1 ft)3 = 6.39 in3 Its crucial that your place the power in the correct position. 12 in is equal to 1 ft, but 1728 in3 and equal to 1 ft3.

31. Convert 4.75 x 107 ft² into mi².

(4.75 * 107 ft3)(1 mi/5280 ft)2 = 1.70 mi2

32. Convert 3.6 m³ into in³.

covert meter3-->cm3-->inch3 (3.6 m3)(100 cm/1 m)3(1 in/2.54 cm)3 = 2.20 * 105 in3

The next problems get complex, but they are conceptually no more difficult than what you've been doing. Try to follow the steps in this example. Convert 3.08 lb/in2 into nt/cm2 solution: First I must explain that nt is for newton, a metric unit of force. It converts directly to pounds by the factor 1 pound = 4.45 newtons. (The correct international system symbol for newton is N, but that can get confused with another N when you're first learning the subject.) 1) Start with your quantity: 3.08 ln/in2 2) Throw in some parentheses: (3.08 lb/in2)(______)(______) 3) Convert pounds to newtons: (3.08 lb/in2)(4.45 nt/1 lb)(______) 4) Convert in2 to cm2: (3.08 lb/in2)(4.45 nt/1 lb)(1 in/2.54 cm)2 5) Multiply it out: (3.08 lb/in2)(4.45 nt/1 lb)(1 in/2.54 cm)2 = 2.12 nt/cm2 As you can see, a systematic approach will substitute nicely for thinking or understanding on these problems, and will free your mind for more important matters.

33. Convert 5.62 nt-m/sec to lb-ft/sec.

(5.62 nt*m/sec)(1 lb/4.45 nt)(100 cm/1 m)(1 in/2.54 cm)(1 ft/12 in) = 4.14 lb*ft/sec

34. Convert 3.91 nt/liter into pounds/quart.

(3.91 nt/L)(1 lb/4.45 nt)(1 L/1.057 qt) = 0.831 lb/qt

35. Convert 4.6 x 104 nt/m² to lb/in².

(4.6 * 104 nt/m2)(1 lb/4.45 nt)(1 m/100 cm)2(2.54 cm/1 in) 2 = 6.67 lb/in2 Don't forget to square 100 and 2.54!

36. Convert 88 feet/second to mile/hour.

(88 ft/sec)(1 mi/5280 ft)(60 sec/1 min)(60 min/1 hr) = 60 mi/hr A handy conversion: 88 ft/sec = 60 mi/hr

37. Convert 17.8 ft³/min² to cm³/sec².

(17.8 ft3/min2)(12 in/1 ft)3(2.54 cm/1 in)3(1 min/60 sec) 2 = 140 cm3/sec2