In order to solve problems involving distance, it is necessary to
A typical problem involving distance and the formula
d = rt is usually entitled a
uniform motion problem. The problem will have something to do with objects moving at a constant
rate of
speed or an average
rate of speed.
Suppose two sisters live 240 miles apart. One sister has three young children who are planning to visit their aunt for a week. To prevent driving so far, the sisters agree to leave at the same time, drive toward each other, and meet somewhere along the route. The sister with the three children tends to drive carefully and obey the
speed limit. Her average
rate of
speed is 70 mph. The other sister drives too fast, and her average
rate of
speed is 80 mph. How long will it take the two sisters to meet each other to transfer the children?
First, notice that two cars are traveling toward each other. Their average rates of
speed are given. The total distance is given, and, even though we do not know the time the cars began to travel, we are told that the two cars did leave home at the same time.
Two tools are very helpful in writing an
equation for a uniform motion problem: diagrams and charts.
A typical diagram to represent this
data would look like the following:
| One Sister’s Home |
Second Sister’s Home |
|
|
240 miles
|
This indicates that sister one is traveling from her home toward sister two. Sister two, likewise, is traveling from her home toward sister one. Together, they will travel a total distance of 240 miles - the distance from one home to the other.
A
chart would be used to include all the
data necessary for writing an equation. The
chart (shown below) uses the formula
distance = rate times time (d = rt). In the
chart below, sister one is the sister with the children who drives at 70 miles per hour, and sister two is the sister driving to pick up the children. She drives 80 miles per hour. The speeds are indicated in the
rate column. The amount of time they travel is unknown and is designated by the letter
t in the time column. The distance column contains the
product of the
rate and the time.
| |
|
time
|
distance
|
|
Sister 1
|
70 mph |
t
|
70t
|
|
Sister 2
|
80 mph |
t
|
80t
|
|
total distance =
|
240 miles |
Using the
chart and the diagram, we now write the equation. The last column of the
chart tells us the total distance for each sister, and the diagram tells us the total distance driven by the two sisters together. Combining these two ideas we have:
| (distance of Sister 1) |
plus |
(distance of Sister 2) |
equals |
(total distance) |
|
70t
|
+
|
80t
|
=
|
240
|
We are now ready to solve the equation:
The result indicates that it will take 1 and
hour for the two sisters to meet. Therefore, it will take 1 hour and 36 minutes for them to meet.