My prescription for minimizing sloppy errors when doing problems:
- Draw a large diagram. Don't worry about trees: in the long run you'll save paper by doing the problem correctly the first time.
- Define a unique variable for everything in the problem. Label as many parts of the diagram as possible.
- Write down the value of every variable you're given. But keep using the variable – not its numerical value – anyway. Keep a mental inventory of which variables are known and unknown.
- Based on the relevant physics, write down a system of equations using variables.
- Make sure all the relevant information in the problem appears in an equation somewhere.
- Check if # equations = # unknowns.
- If so, the physics is done: everything else is just algebra. If not, it's possible that one of the unknowns is irrelevant and will cancel out in the end: proceed with caution.
- Sketch out how you will solve the system of equations. A little forethought will prevent you from endless loops of substituting one equation into another and back again.
- In a homework problem you'll probably have to work through to the bitter end.
- On an exam, I would give you 80-90% credit for reaching this point. You may want to skip actually doing the algebra: come back to it later if you have time.
- If you're just studying, stop here. We all know you could do the algebra if necessary, but it's probably not going to add to your understanding of the physics.
- Solve for whatever the problem asks about (which your book calls the "target variable") algebraically using variables. You will end up with a more-or-less succinct expression with which you can
- Check units. You don't even need to put numbers in: just make sure the units of the final expression match the units of the target variable. If not, you probably made an algebra mistake. You can check units of intermediate results to find the problem.
- Plug in numbers. This is the first time you ever use the value of a variable.
- Check that the result is reasonable. You probably have an idea of how big the answer should be. If you're way off, 9 times out of 10 you just keyed in a value incorrectly.
This procedure does not guarantee success, but it makes it much more likely that you will translate an understanding of the physics into a correct final answer with minimal pounding on the calculator.