The basic formula for motion problems is

( *r*) × ( *t*) = *d* [or *d* = ( *r*) × ( *t*)]

(rate) times (time) = distance [or distance equals (rate) times (time)]

Example 1

A train leaves Chicago at 11:00 a.m. traveling east at a speed of 40 mph. Two hours later, a second train leaves Chicago on a parallel track traveling in the same direction as the first train. The second train travels at a rate of 50 mph. Assuming that neither train stops, at what time will the second train catch up to the first train? At that time, how far has each train traveled?

Drawing a diagram (see Figure 1) helps in understanding the situation. Constructing a chart helps in organizing the data.

Since both trains began at the same point and will be at the same point when the second train catches up to the first, they will have traveled the same distance. Let *t* equal the number of hours the first train travels. Since the second train leaves two hours later, it will have traveled for two hours less. Therefore, *t* – 2 equals the number of hours the second train travels (see Figure 2).

The trains travel the same distance, so

Therefore, it will be 10 hours after the first train leaves before the second train catches the first train. *But this is not the question!* The question asks, “At what time will the second train catch up to the first train?” The second train will catch up to the first train 10 hours after 11:00 a.m., which is 9:00 p.m. The second question asks, “At that time, how far has each train traveled?” Replace *t* into either distance category and evaluate.

40 *t* = 40(10) = 400 or 50( *t* – 2) = 50(10 – 2) = 400

Therefore, each train has traveled 400 miles. Notice that the answer to the algebra is not always the answer to the question.

Figure 1. A diagram for the problem posed in Example.

Figure 2. Working through Example.

Example 2

Susan's boat can go 9 mph in still water. She can go 44 miles downstream in a river in the same time as it would take her to go 28 miles upstream. What is the speed of the river? Figure 3 shows this situation.

When Susan goes in the same direction as the river, her speed increases by the speed of the river. When she goes against the river, upstream, her speed decreases by the speed of the river.

Organize the information in a chart (see Figure 4).

Since *(r)(t) = d*, then

In the problem, the key phrase is “in the same time.” So set the times *T* _{1} and *T* _{2} equal.

Therefore, the river's speed is 2 mph. The check is left to you.

Figure 3. A current event, of sorts.

Figure 4. A chart for the problem in Example.