Neatness and organization. I am not your mother,
and I will not tell you how to organize either your dorm room or
your problem solutions. But I can tell you that it is easier to
work from neat, well-organized pages than from scribbles. I can
also warn you about certain handwriting pitfalls: Distinguish
carefully between t and +, between l and 1, and between Z and 2.
(I write a t with a hook at the bottom, an l in script lettering,
and a Z with a cross bar. You can form your own conventions.)
These suggestions on neatness, organization, and handwriting do
not arise from prudishness--they are practical suggestions that
help avoid algebraic errors, and they are for your benefit, not
mine. (On the other hand, it doesn't hurt to be neat and
organized for the benefit of your grader. One course grader of
mine pointed out: "If I can't read it, I can't give you
credit.")
Avoid needless conversions. If the problem gives you one length in meters and another in inches, then it's probably best to convert all lengths to meters. But if all the lengths are in inches, then there's no need to convert everything to meters--your answer should be in inches. In fact, you might not actually need to convert. For example, perhaps two lengths are given in inches and the final answer turns out to depend only on the ratio of those two lengths. In that case, the ratio is the same whether the lengths going into the ratio are inches or meters. It's easy to make arithmetic errors while doing conversions. If you don't convert, then you don't make those errors!
Keep it simple. I will not assign baroque problems that require tortuous explanations and pages of algebra. If you find yourself working in such a way, then you're on the wrong path. The cure is to stop, go back to the beginning, and start over with a new strategy. (Generations of students have kept track of this rule by remembering to KISS: Keep It Simple and Straightforward.)
Answer Checking. Checking your answer does not mean comparing it to the answer in the back of the book. It means finding the characteristics of your answer and comparing them to the characteristics that you expect. Some of your problems--particularly the ones assigned early in the course--will actually lead you through the checking stage in order to familiarize you with the process. Other problems will leave it to you to perform this check. In either case, checking your answer is not just good problem solving practice that helps you gain points on problem assignments and on exams. The checking stage builds familiarity with the content of physics and the character of problem solutions, and hence develops your intuition to make solving other problems--and learning more physics--easier. (See Daniel F. Styer, "Guest comment: Getting there is half the fun", American Journal of Physics 64 (1998) 105-106.)
Dimensional analysis. Suppose you find a formula for distance (in, say, meters) in terms of some information about velocity (meters/second), acceleration (meters/second2), and time (seconds). If your formula is correct then all of the dimensions on the right hand side must cancel so as to end up with "meters".
Numerical reasonableness. If your problem asks you to find the mass of a squirrel, do you find a mass of 1,970 kilograms? Even worse, do you find a mass of -1,970 kilograms?
[Reasonable speeds. "My calculations give me a speed of 23 m/s. Is this reasonable?" It's hard for most people to get a feel for the reasonableness of speeds expressed in meters per second. Until this qualitative feel develops, Americans should check for reasonableness by converting speeds in meters per second to speeds in miles per hour: simply double the number (20 m/s is about 40 mi/hr). Non-Americans should convert to kilometers per hour: simply quadruple the number (20 m/s is about 80 km/hr).]
Algebraically possible. Would evaluating your formula ever lead you to divide by zero or take the square root of negative number?
Functionally reasonable. Does your answer depend on the given quantities in a reasonable way? For example, you might be asked how far a projectile travels after it is launched at a given speed with a given angle. Common sense says that if the initial speed is increased (keeping the angle constant) then the distance traveled will increase. Does your formula agree with common sense?
Limiting values and special cases. In the projectile travel distance problem mentioned above, the range is obviously zero for a vertical launch. Does your formula give this result? If you solve a problem regarding two objects, does it give the proper result when the two objects have equal masses? When one of them has zero mass (i.e. does not exist)?
Symmetry. Problems often have geometrical symmetry from which you can determine the direction of a vector but not its magnitude. More often they have a "permutation" symmetry: If your problem has two objects, you can call the cube "object number 1" and the sphere "object number 2" but your final answer had better not depend upon how you numbered your objects. (That is, it should give the same answer if every "1" is changed to a "2" and vice versa.)
Specify units. "The distance is 5.72" is not an answer. Is that 5.72 miles, 5.72 meters, or 5.72 inches? Similarly, if the answer is a vector, both magnitude and direction must be specified. (The direction may be drawn into a diagram rather than stated explicitly.)
Significant figures. Any number that comes from an experiment comes with some uncertainty. Most of the numbers in this course come with three significant figures. If a ball rolls 3.24 meters in 2.41 seconds, then report its speed as 1.34 m/s, not 1.34439834 m/s. Most introductory physics courses do not require a formal or technical error analysis, but you should avoid inaccurate statements like the second quotient above.
Large problems. If you break up your large problem into several sub-problems, as recommended above, then check your results at the end of each sub-problem. If your answer to the second sub-problem passes its checks, but your answer to the third sub-problem fails its checks, then your execution error almost certainly falls within the third sub-problem. Knowing its general location, you can quickly go back and correct the error, so its effects will not propagate on to the remaining sub-problems. This can be a real time-saver.
Summary
The problems in your physics course can be fun and exciting. Approach them in the spirit of exploration and they will not disappoint you!
Strategy design
Classify the problem by its method of solution.
Summarize the situation with a diagram.
Keep the goal in sight (perhaps by writing it down).
Execution tactics
Work with symbols.
Keep packets of related variables together.
Be neat and organized.
Keep it simple.
Answer checking
Dimensionally consistent?
Numerically reasonable (including sign)?
Algebraically possible? (Example: no imaginary or infinite answers.)
Functionally reasonable? (Example: greater range with greater initial speed.)
Check special cases and symmetry.
Report numbers with units specified and with reasonable significant figures.
Image Credit: fallout
Avoid needless conversions. If the problem gives you one length in meters and another in inches, then it's probably best to convert all lengths to meters. But if all the lengths are in inches, then there's no need to convert everything to meters--your answer should be in inches. In fact, you might not actually need to convert. For example, perhaps two lengths are given in inches and the final answer turns out to depend only on the ratio of those two lengths. In that case, the ratio is the same whether the lengths going into the ratio are inches or meters. It's easy to make arithmetic errors while doing conversions. If you don't convert, then you don't make those errors!
Keep it simple. I will not assign baroque problems that require tortuous explanations and pages of algebra. If you find yourself working in such a way, then you're on the wrong path. The cure is to stop, go back to the beginning, and start over with a new strategy. (Generations of students have kept track of this rule by remembering to KISS: Keep It Simple and Straightforward.)
Answer Checking. Checking your answer does not mean comparing it to the answer in the back of the book. It means finding the characteristics of your answer and comparing them to the characteristics that you expect. Some of your problems--particularly the ones assigned early in the course--will actually lead you through the checking stage in order to familiarize you with the process. Other problems will leave it to you to perform this check. In either case, checking your answer is not just good problem solving practice that helps you gain points on problem assignments and on exams. The checking stage builds familiarity with the content of physics and the character of problem solutions, and hence develops your intuition to make solving other problems--and learning more physics--easier. (See Daniel F. Styer, "Guest comment: Getting there is half the fun", American Journal of Physics 64 (1998) 105-106.)
Dimensional analysis. Suppose you find a formula for distance (in, say, meters) in terms of some information about velocity (meters/second), acceleration (meters/second2), and time (seconds). If your formula is correct then all of the dimensions on the right hand side must cancel so as to end up with "meters".
Numerical reasonableness. If your problem asks you to find the mass of a squirrel, do you find a mass of 1,970 kilograms? Even worse, do you find a mass of -1,970 kilograms?
[Reasonable speeds. "My calculations give me a speed of 23 m/s. Is this reasonable?" It's hard for most people to get a feel for the reasonableness of speeds expressed in meters per second. Until this qualitative feel develops, Americans should check for reasonableness by converting speeds in meters per second to speeds in miles per hour: simply double the number (20 m/s is about 40 mi/hr). Non-Americans should convert to kilometers per hour: simply quadruple the number (20 m/s is about 80 km/hr).]
Algebraically possible. Would evaluating your formula ever lead you to divide by zero or take the square root of negative number?
Functionally reasonable. Does your answer depend on the given quantities in a reasonable way? For example, you might be asked how far a projectile travels after it is launched at a given speed with a given angle. Common sense says that if the initial speed is increased (keeping the angle constant) then the distance traveled will increase. Does your formula agree with common sense?
Limiting values and special cases. In the projectile travel distance problem mentioned above, the range is obviously zero for a vertical launch. Does your formula give this result? If you solve a problem regarding two objects, does it give the proper result when the two objects have equal masses? When one of them has zero mass (i.e. does not exist)?
Symmetry. Problems often have geometrical symmetry from which you can determine the direction of a vector but not its magnitude. More often they have a "permutation" symmetry: If your problem has two objects, you can call the cube "object number 1" and the sphere "object number 2" but your final answer had better not depend upon how you numbered your objects. (That is, it should give the same answer if every "1" is changed to a "2" and vice versa.)
Specify units. "The distance is 5.72" is not an answer. Is that 5.72 miles, 5.72 meters, or 5.72 inches? Similarly, if the answer is a vector, both magnitude and direction must be specified. (The direction may be drawn into a diagram rather than stated explicitly.)
Significant figures. Any number that comes from an experiment comes with some uncertainty. Most of the numbers in this course come with three significant figures. If a ball rolls 3.24 meters in 2.41 seconds, then report its speed as 1.34 m/s, not 1.34439834 m/s. Most introductory physics courses do not require a formal or technical error analysis, but you should avoid inaccurate statements like the second quotient above.
Large problems. If you break up your large problem into several sub-problems, as recommended above, then check your results at the end of each sub-problem. If your answer to the second sub-problem passes its checks, but your answer to the third sub-problem fails its checks, then your execution error almost certainly falls within the third sub-problem. Knowing its general location, you can quickly go back and correct the error, so its effects will not propagate on to the remaining sub-problems. This can be a real time-saver.
Summary
The problems in your physics course can be fun and exciting. Approach them in the spirit of exploration and they will not disappoint you!
Strategy design
Classify the problem by its method of solution.
Summarize the situation with a diagram.
Keep the goal in sight (perhaps by writing it down).
Execution tactics
Work with symbols.
Keep packets of related variables together.
Be neat and organized.
Keep it simple.
Answer checking
Dimensionally consistent?
Numerically reasonable (including sign)?
Algebraically possible? (Example: no imaginary or infinite answers.)
Functionally reasonable? (Example: greater range with greater initial speed.)
Check special cases and symmetry.
Report numbers with units specified and with reasonable significant figures.
Image Credit: fallout
