1.7.2 Drawing Pictures
You may have been told that the first step in solving a physics problem is to “draw a picture,” but perhaps you didn’t know why, or what to draw. The purpose of drawing a picture is to aid you in the words-to-symbols translation. Complex problems have far more information than you can keep in your head at one time. Think of a picture as a “memory extension,” helping you organize and keep track of vital information.
Although any picture is better than none, there really is a method for drawing pictures that will help you be a better problem solver. It is called the pictorial representation of the problem. We’ll add other pictorial representations as we go along, but the following procedure is appropriate for motion problems.
TACTICS BOX 1.4 Drawing a pictorial representation
1 Draw a motion diagram. The motion diagram develops your intuition for the motion.
2 Establish a coordinate system. Select your axes and origin to match the motion. For one-dimensional motion, you want either the x-axis or the y-axis parallel to the motion. The coordinate system determines whether the signs of v and a are positive or negative.
3 Sketch the situation. Not just any sketch. Show the object at the beginning of the motion, at the end, and at any point where the character of the motion changes. Show the object, not just a dot, but very simple drawings are adequate.
4 Define symbols. Use the sketch to define symbols representing quantities such as position, velocity, acceleration, and time. Every variable used later in the mathematical solution should be defined on the sketch. Some will have known values, others are initially unknown, but all should be given symbolic names.
5 List known information. Make a table of the quantities whose values you can determine from the problem statement or that can be found quickly with simple geometry or unit conversions. Some quantities are implied by the problem, rather than explicitly given. Others are determined by your choice of coordinate system.
6 Identify the desired unknowns. What quantity or quantities will allow you to answer the question? These should have been defined as symbols in step 4. Don’t list every unknown, only the one or two needed to answer the question.
It’s not an overstatement to say that a well-done pictorial representation of the problem will take you halfway to the solution. The following example illustrates how to construct a pictorial representation for a problem that is typical of problems you will see in the next few chapters.
EXAMPLE 1.8 Drawing a pictorial representation
Draw a pictorial representation for the following problem: A rocket sled accelerates horizontally at for then coasts for . What is the total distance traveled?
VISUALIZE FIGURE 1.20 is the pictorial representation. The motion diagram shows an acceleration phase followed by a coasting phase. Because the motion is horizontal, the appropriate coordinate system is an x-axis. We’ve chosen to place the origin at the starting point. The motion has a beginning, an end, and a point where the motion changes from accelerating to coasting, and these are the three sled positions sketched in the figure. The quantities x, , and t are needed at each of three points, so these have been defined on the sketch and distinguished by subscripts. Accelerations are associated with intervals between the points, so only two accelerations are defined. Values for three quantities are given in the problem statement, although we need to use the motion diagram, where we find that points to the right, to know that rather than . The values and are choices we made when setting up the coordinate system. The value is part of our interpretation of the problem. Finally, we identify as the quantity that will answer the question. We now understand quite a bit about the problem and would be ready to start a quantitative analysis.
FIGURE 1.20 A pictorial representation.
We didn’t solve the problem; that is not the purpose of the pictorial representation. The pictorial representation is a systematic way to go about interpreting a problem and getting ready for a mathematical solution. Although this is a simple problem, and you probably know how to solve it if you’ve taken physics before, you will soon be faced with much more challenging problems. Learning good problem-solving skills at the beginning, while the problems are easy, will make them second nature later when you really need them.