🚩 Problem Solving

Problem Solving

Solving problems is the most common task used to measure understanding in technical and scientific courses, and in many aspects of life as well. In general, problem solving requires factual and procedural knowledge in the area of the problem, plus knowledge of numerous schema, plus skill in overall problem solving. Schema is loosely defined as a “specific type of problem” such as principal, rate, and interest problems, one-dimensional kinematic problems with constant acceleration, etc. In most introductory courses, improving problem solving relies on three things:

1. increasing domain knowledge, particularly definitions and procedures
2. learning schema for various types of problems and how to recognize that a particular problem belongs to a known schema
3. becoming more conscious of and insightful about the process of problem solving.

To improve your problem solving ability in a course, the most essential change of attitude is to focus more on the process of solution rather than on obtaining the answer. For homework problems there is frequently a simple way to obtain the answer, often involving some specific insight. This will quickly get you the answer, but you will not build schema that will help solve related problems further down the road. Moreover, if you rely on insight, when you get stuck on a problem, you’re stuck with no plan or fallback position.

General Approach to Problem Solving

A great many physics textbook authors recommend overall problem solving strategies. These are typically four-step procedures that descend from George Polya’s influential book, How to Solve It, on problem solving. Here are his four steps:

1. Understand – get a conceptual grasp of the problem What is the problem asking? What are the given conditions and assumptions? What domain of knowledge is involved? What is to be found and how is this determined or constrained by the given conditions? What knowledge is relevant? E.g. in physics, does this problem involve kinematics, forces, energy, momentum, angular momentum, equilibrium? If the problem involves two different areas of knowledge, try to separate the problem into parts. Is there motion or is it static? If the problem involves vector quantities such as velocity or momentum, think of these geometrically (as arrows that add vectorially). Get conceptual understanding: is some physical quantity (energy, momentum, angular momentum, etc.) constant? Have you done problems that involve the same concepts in roughly the same way?

   Model: Real life contains great complexity, so in physics (chemistry, economics …) you actually solve a model problem that contains the essential elements of the real problem. The bike and rider become a point mass (unless angular momentum is involved), the ladder’s mass is regarded as being uniformly distributed along its length, the car is assumed to have constant acceleration or constant power (obviously not true when it shifts gears), etc. Become sensitive to information that is implicitly assumed (Presence of gravity? No friction? That the collision is of short duration relative to the timescale of the subsequent motion? …).

   Advice: Write your own representation of the problem’s stated data; redraw the picture with your labeling and comments. Get the problem into your brain! Go systematically down the list of topics in the course or for that week if you are stuck.

2. Devise a Plan - set up a procedure to obtain the desired solution

   General - Have you seen a problem like this – i.e., does the problem fit in a schema you already know? Is a part of this problem a known schema? Could you simplify this problem so that it is? Can you find any useful results for the given problem and data even if it is not the solution (e.g. in the special case of motion on an incline when the plane is at \(\theta = 0\))? Can you imagine a route to the solution if only you knew some apparently not given information? If your solution plan involves equations, count the unknowns and check that you have that many independent equations.

   In Physics, exploit the freedoms you have: use a particular type of coordinate system (e.g. polar) to simplify the problem, pick the orientation of a coordinate system to get the unknowns in one equation only (e.g. only the \(x\)-direction), pick the position of the origin to eliminate torques from forces you don’t know, pick a system so that an unknown force acts entirely within it and hence does not change the system’s momentum … Given that the problem involves some particular thing (constant acceleration, momentum) think over all the equations that involve this concept.

3. Carry our your plan – solve the problem! This generally involves mathematical manipulations. Try to keep them as simple as possible by not substituting in lengthy algebraic expressions until the end is in sight, make your work as neat as you can to ease checking and reduce careless mistakes. Keep a clear idea of where you are going and have been (label the equations and what you have now found), if possible, check each step as you proceed. Always check dimensions if analytic, and units if numerical.

4. Look Back – check your solution and method of solution Can you see that the answer is correct now that you have it – often simply by retrospective inspection? Can you solve it a different way? Is the problem equivalent to one you’ve solved before if the variables have some specific values? Check special cases (for instance, for a problem involving two massive objects moving on an inclined plane, if \(m_1 = m_2\) or \(\theta = 0\) does the solution reduce to a simple expression that you can easily derive by inspection or a simple argument?) Is the scaling what you’d expect (an energy should vary as the velocity squared, or linearly with the height). Does it depend sensibly on the various quantities (e.g. is the acceleration less if the masses are larger, more if the spring has a larger \(k\) )? Is the answer physically reasonable (especially if numbers are given or reasonable ones substituted).

Review the schema of your solution: Review and try to remember the outline of the solution – what is the model, the physical approximations, the concepts needed, and any tricky math manipulation.