MODEL Here is a problem where, for the first time, we can really put all four aspects of our problem-solving strategy into play. To begin, we’ll model Bob’s and Susan’s cars as being in uniform motion. Their real motion is certainly more complex, but over a long drive it’s reasonable to approximate their motion as constant speed along a straight line.

VISUALIZE FIGURE 2.5 shows the pictorial representation. The equal spacings of the dots in the motion diagram indicate that the motion is uniform. In evaluating the given information, we recognize that the starting time of 9:00 a.m. is not relevant to the problem. Consequently, the initial time is chosen as simply $t_0=0\text{h}$. Bob and Susan are traveling in opposite directions, hence one of the velocities must be a negative number. We have chosen a coordinate system in which Bob starts at the origin and moves to the right (east) while Susan is moving to the left (west). Thus Susan has the negative velocity. Notice how we’ve assigned position, velocity, and time symbols to each point in the motion. Pay special attention to how subscripts are used to distinguish different points in the problem and to distinguish Bob’s symbols from Susan’s.

FIGURE 2.5 Pictorial representation for Example 2.2.

One purpose of the pictorial representation is to establish what we need to find. Bob and Susan meet when they have the same position at the same time $t_1$. Thus we want to find ${\left(x_1\right)}_{\text{B}}$ at the time when ${\left(x_1\right)}_{\text{B}}=(x_1){}_{\text{S}}$. Notice that ${\left(x_1\right)}_{\text{B}}$ and ${\left(x_1\right)}_{\text{S}}$ are Bob’s and Susan’s positions, which are equal when they meet, not the distances they have traveled.

SOLVE The goal of the mathematical representation is to proceed from the pictorial representation to a mathematical solution of the problem. We can begin by using Equation 2.3 to find Bob’s and Susan’s positions at time $t_1$ when they meet:

Notice two things. First, we started by writing the full statement of Equation 2.3. Only then did we simplify by dropping those terms known to be zero. You’re less likely to make accidental errors if you follow this procedure. Second, we replaced the generic symbol $s$ with the specific horizontal-position symbol x, and we replaced the generic subscripts i and f with the specific symbols 0 and 1 that we defined in the pictorial representation. This is also good problem-solving technique.

The condition that Bob and Susan meet is

By equating the right-hand sides of the above equations, we get

Solving for $t_1$ we find that they meet at time

Finally, inserting this time back into the equation for ${\left(x_1\right)}_{\text{B}}$ gives

As noted in Chapter 1, this textbook will assume that all data are good to at least two significant figures, even when one of those is a trailing zero. So 400 miles, 60 mph, and 40 mph each have two significant figures, and consequently we’ve calculated results to two significant figures.

While 240 miles is a number, it is not yet the answer to the question. The phrase “240 miles” by itself does not say anything meaningful. Because this is the value of Bob’s position, and Bob was driving east, the answer to the question is, “They meet 240 miles east of Chicago.”

REVIEW Before stopping, we should check whether or not this answer seems reasonable. We certainly expected an answer between 0 miles and 400 miles. We also know that Bob is driving faster than Susan, so we expect that their meeting point will be more than halfway from Chicago to Pittsburgh. Our review tells us that 240 miles is a reasonable answer.