Journal of Online Mathematics and its Applications
While the focus of these modules in Math and Physics is on conceptual learning, a great deal of the "utility" of introductory courses in calculus and physics is in developing problem-solving skills. Hence any textbook has many traditional examples of problems and many more exercises for the student than are presented within these modules.
A proper introduction to problem-solving starts by being reminded that the essence of scientific progress since Galileo is in experimenation and experimentation is founded on quantitative measurement.
As an example, consider the pictures below. They show two scenes of a girl walking a cow to pasture while simultaneously holding a kite on a windy day. Which of the two scenes are you more likely to see on a March day in Pennsylvania farmland?
Hard as it is to believe, the general principles of physics embodied in Newton's Laws do not, of themselves, preclude either scene from happening. These laws, when coupled with physical intuition as to the masses of objects, do give us a firm indication as to the likelihood of the second scene compared to the first. With that in mind, we can start to look from our qualitative view of how things should move to the techniques for considering quantitative determinations of their motion.
Published July 2001
© 2001 by Larry Gladney and Dennis DeTurck
Units and Dimensional Analysis
To Measure All Things
Modern science begins with observation and description. The appropriate means of description should allow for little ambiguity in comparison of observations from different people. In many ways, science is more of a driver for a ``universal'' language than almost any human activity. Although mathematics suffices as a universal language for theoretical purposes, when it comes to experimental measurement, we turn to a globally agreed upon set of standards termed SI or Système International.
The SI units for length, mass, and time are the
Combinations of these basic units are used to make up other SI units for quantifying energy, power, speed, force, etc. The SI units all make use of the same set of prefixes for specifying appropriate powers of ten of the basic unit. For example, the centimeter is 1/100 of a meter, a kilometer is 1000 meters, etc. The list of accepted prefixes is below.
All calculations you do for this class should be in terms of SI units. It will not always be necessary to use SI units to get the correct answer, but all of the formulas we use assume that you are using SI units and are defined with that in mind.
Conversion Between UnitsDealing with units can be annoying when it is necessary to transform from one set to another. You will find below a handy interactive conversion table for doing just that.
In terms of operation, we generally will need to simply multiply through by appropriate conversion factors in a calculation to get to the standard units, for example, suppose we need to find the speed of a car in m/s for a particular problem, but the statement of the problem gives the speed as 30 miles/hour. Then, we perform the following operation by getting the conversion factors from the unit conversion program, then
Another useful reason for keeping track of units is that they generally give the first hint that a numerical error may have occurred in your calculations. Generally, converting from one unit to another, a technique termed dimensional analysis, is a necessary part of the calculation of an answer to a problem, hence if the units do not come out correctly, you can be sure that there is something not right with your solution. Another thing to watch for is reasonableness of the answer. Physics classes generally provide problems with answers that are actually realizable in real life. Cars do not travel 8000 miles/hour, people can't apply 4 million pounds of force to a door. If the numbers don't look like they fit the size of the unit, check your calculation!
If you haven't done it in awhile, you will need to practice a bit with unit conversions. Let's try the following exercises, using the conversion tables that you can access with the following link.
Dimensions of Physical Quantities
When dealing with calculations of physical quantities, we need to be careful to not add "apples and oranges". The following table gives the dimensions for various physical quantities with L standing for a dimension of length, m for mass, and t for time. We will come across all of these in this course of study.
In order to do any mathematical operation (add, divide, multiply, or subtract) with any of these physical quantities, we must be sure that all the physical quantities involved in the operation are of the same type. For example, it is wrong to do the operation
if A is a velocity and B is a force. The dimensions for C would be undefined. Many great ideas in physics were begun on the basis of making calculations that were dimensionally correct. For example, physicists had noted that two of the fundamental constants in physics that determine the strength of electric and magnetic interactions, when multiplied gave units which were the inverse square of a velocity. The precise number was exactly equal to the inverse square of the speed of light in fact. This was the first hint of a deep connection between what appeared to be very different physical phenomena: electricity, magnetism, and optics.
Dimensional analysis is also one of the methods for determining whether some error has occurred in the midst of a calculation since units must be consistent at all points.
Significant Figures and Scientific Notation
All physical measurements are limited by intrinsic uncertainties in the method by which they are made. Even in the ideal exact measurement is not possible since ultimately quantum mechanics imposes fundamental limits on what we can quantitatively know about a system. Since the measurement technique determines the uncertainty in the measurement, quantitative sciences ONLY deal with measurements which are accompanied by the associated uncertainty. Hence it is never appropriate to state the result of a laboratory measurement as, say 3 centimeters. We are required to state the uncertainty as part of the measurement, e.g. 3 ± 0.2 cm. The 0.2 cm indicates that we can have made an error of as much 0.2 cm in the measurement of 3 cm so someone else making the measurement with a similar or more accurate instrument might get a value as low as 2.8 cm (but not likely a lower value) or a number as high as 3.2 cm (but not likely a higher value).
In making calculations, we need to keep in mind that the significant figures are determined by the last digit (other than the zeroes used to locate the decimal point) which is believed reliable. Hence, if our measurement of a speed is accurate to within 5 mm/s then we would state a measurement of speed in meters per second with 2 decimal places after the decimal point as in 4.52 m/s. A third decimal place after the decimal point would not be reliable as it could be off by as much as ± 5. The number of significant digits is determined by the number of digits which are reliable, therefore 4.52 has 3 significant figures. A number such as 121.83 has five significant figures and 0.00012 has two significant figures since the zeroes before the "12" only locate the decimal point.
As a matter of convenience for writing and a means of emphasizing the significant figures of a measurement, we use scientific notation for numbers. This notation states numbers as a product of a number and a power of 10. For example, the number 0.00012 in scientific notation would be stated as
We can apply a combination of significant figures, scientific notation and unit conversions to a typical problem.
Example: The SI unit of volume is the liter, defined as the volume that fits in a cube 10 cm along each dimension. What is the density of water (in g/cm3) if 2200 pounds occupies a volume of 1000 liters?
Estimation and Orders of Magnitude
Developing a good sense of how to use your physical intuition for solving problems involves learning the technique of deciding which approach to solving a problem tends to work more "easily" than other possible approaches. In some cases, detailed mathematical analysis is necessary, but, for many questions, an amazing degree of accuracy can be obtained by a series of "educated guesses" for which the correct order of magnitude for each quantity involved and simple calculations are all is all that is necessary to get a surprisingly accurate answer. The great nuclear physics Enrico Fermi is acknowledged as one of the great masters of this technique and used it to find the answers to what would seem to be almost impossible questions to answer. Hence the kind of problems that can be attacked in this way are termed Fermi problems. Here are some examples:
Although our answer may be off by a factor of 2 or 3, it is almost certainly not off by a factor of 10. For some problems this is more than adequate.
So 30,000 people can fit onto the field itself.
Sometimes you can use the Fermi method to get some amazingly accurate results from your everyday knowledge. For example, suppose you are asked to state the radius of the earth and you do not have that quantity handily available. Think about your (or others) travel experiences:
The actual answer is 6.37 x 106 m so you can see that the Fermi estimate gets remarkably close.
Although the breadth of the Internet and smart search engines have made it possible to quickly find incredible amounts of information, it is still useful to keep in mind some orders of magnitude for how big or small things are for just the kind of quick Fermi problem calculations we might use to decide how to attack particular problems. Here are a few quantities that might prove useful along with the Web links to where the information was derived. You may wish to explore these links at your leisure, but it is not necessary to do so. These links were valid as of Jun 1, 2001. [Editor's note: Links checked Nov 8, 2004.]