In addition to text and interactive problem sections, this textbook contains sections with sample problems and derivations of equations. Sample problems often demonstrate a useful problem-solving technique. You see a typical sample problem above. Derivations show how an equation new to you can be created from equations you have already learned.

We follow the same sequence of steps in sample problems and derivations. (You will also follow this same sequence when you work through problems called interactive checkpoints; more on this type of problem in the next section.) Sample problems, derivations and interactive checkpoints all have some or all of the following: a diagram, a table of variables, a statement of the problem-solving strategy, the principles and equations used, and a step-by-step solution.

To show how these are organized, we work through a sample problem from the study of linear motion. The problem is stated above.

Draw a diagram

It is often helpful to draw a diagram of the problem, with important values labeled. Although almost every problem is stated using an illustration, we sometimes find it useful to draw an additional diagram.

Variables

We summarize the variables relating to the problem in a table. Some of these have values given in the problem statement or illustration. If we do not know the value of a variable, we enter the variable symbol. A variable table for the problem stated above is shown.

There are two reasons we write the variables. One is so that if you see a variable with which you are unfamiliar, you can quickly see what it represents. The other is that it is another useful problem-solving technique: Write down everything you know. Sometimes you know more than you think you know! Some variables may also prompt you to think of ways to solve the problem.

After these two steps, we move to strategy.

What is the strategy?

The strategy is a summary of the sequence of steps we will follow in solving the problem. Some students who used this book early in its development called the strategy section “the hints,” which is another way to think of the strategy. There are typically many ways to solve a problem; our strategy is the one we chose to employ. (As we point out in the text when we actually solve this problem, there is another efficient manner in which to solve it.)

For the problem above, our strategy was:

1. There are two unknowns, the initial and final velocities, so choose two equations that include these two unknowns and the values you do know.
2. Substitute known values and use algebra to reduce the two equations to one equation with a single unknown value.

Principles and equations

Principles and equations from physics and mathematics are often used to solve a problem. For the problem above, for example, these two linear motion equations that apply when acceleration is constant are useful:

v_f = v_i + at

Δx = ½(v_i + v_f)t

The physics principles are the crucial points that the problems are attempting to reinforce. If they look quite familiar to you at some point: Great!

Step-by-step solution

We solve the problem (or work through the derivation) in a series of steps. We provide a reason for each step. If you want a more detailed explanation, you can click on a step, which causes a more detailed text explanation to appear on the right. Some students find the additional information quite helpful; others prefer the very brief explanation. It also varies depending on the difficulty of the problem − everyone can use a little help sometimes.

Here are the first three steps that we used to solve the problem above.