1.7.4 A Problem-Solving Strategy

One of the goals of this textbook is to help you learn a strategy for solving physics problems. The purpose of a strategy is to guide you in the right direction with minimal wasted effort. The four-part problem-solving strategy—Model, Visualize, Solve, Review—is based on using different representations of knowledge. You will see this problem-solving strategy used consistently in the worked examples throughout this textbook, and you should endeavor to apply it to your own problem solving.

GENERAL PROBLEM-SOLVING STRATEGY

MODEL It’s impossible to treat every detail of a situation. Simplify the situation with a model that captures the essential features. For example, the object in a mechanics problem is often represented as a particle.

VISUALIZE This is where expert problem solvers put most of their effort.

Draw a pictorial representation. This helps you visualize important aspects of the physics and assess the information you are given. It starts the process of translating the problem into symbols.
Use a graphical representation if it is appropriate for the problem.
Go back and forth between these representations; they need not be done in any particular order.

SOLVE Only after modeling and visualizing are complete is it time to develop a mathematical representation with specific equations that must be solved. All symbols used here should have been defined in the pictorial representation.

REVIEW Is your result believable? Does it have proper units? Does it make sense?

Throughout this textbook we will emphasize the first two steps. They are the physics of the problem, as opposed to the mathematics of solving the resulting equations. This is not to say that those mathematical operations are always easy—in many cases they are not. But our primary goal is to understand the physics.

Textbook illustrations are obviously more sophisticated than what you would draw on your own paper. To show you a figure very much like what you should draw, the final example of this section is in a “pencil sketch” style. We will include one or more pencil-sketch examples in nearly every chapter to illustrate exactly what a good problem solver would draw.

EXAMPLE 1.9 Launching a weather rocket

Use the first two steps of the problem-solving strategy to analyze the following problem: A small rocket, such as those used for meteorological measurements of the atmosphere, is launched vertically with an acceleration of $30 {m/s}^{2}$ . It runs out of fuel after $30 s$ . What is its maximum altitude?

MODEL We need to do some interpretation. Common sense tells us that the rocket does not stop the instant it runs out of fuel. Instead, it continues upward, while slowing, until it reaches its maximum altitude. This second half of the motion, after running out of fuel, is like the ball that was tossed upward in the first half of Example 1.6. Because the problem does not ask about the rocket’s descent, we conclude that the problem ends at the point of maximum altitude. We’ll model the rocket as a particle.

VISUALIZE FIGURE 1.21 shows the pictorial representation in pencil-sketch style. The rocket is speeding up during the first half of the motion, so $\vec{a}_{0}$ points upward, in the positive y-direction. Thus the initial acceleration is $a_{0 y} = 30 {m/s}^{2}$ . During the second half, as the rocket slows, $\vec{a}_{1}$ points downward. Thus $a_{1 y}$ is a negative number.

A pictorial representation in pencil-sketch style of a weather rocket launch.

FIGURE 1.21 Full Alternative Text

This information is included with the known information. Although the velocity $v_{2 y}$ wasn’t given in the problem statement, it must—just like for the ball in Example 1.6—be zero at the very top of the trajectory. Last, we have identified $y_{2}$ as the desired unknown. This, of course, is not the only unknown in the problem, but it is the one we are specifically asked to find.

REVIEW If you’ve had a previous physics class, you may be tempted to assign $a_{1 y}$ the value $- 9.8 {m/s}^{2}$ , the free-fall acceleration. However, that would be true only if there is no air resistance on the rocket. We will need to consider the forces acting on the rocket during the second half of its motion before we can determine a value for $a_{1 y}$ . For now, all that we can safely conclude is that $a_{1 y}$ is negative.

Our task in this chapter is not to solve problems—all that in due time—but to focus on what is happening in a problem. In other words, to make the translation from words to symbols in preparation for subsequent mathematical analysis. Modeling and the pictorial representation will be our most important tools.