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3.5 Solve uniform motion applications

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By the end of this section, you will be able to:
  • Solve uniform motion applications

Before you get started, take this readiness quiz.

  1. Find the distance travelled by a car going 70 miles per hour for 3 hours.
    If you missed this problem, review [link] .
  2. Solve x + 1.2 ( x − 10 ) = 98 .
    If you missed this problem, review [link] .
  3. Convert 90 minutes to hours.
    If you missed this problem, review [link] .

Solve uniform motion applications

When planning a road trip, it often helps to know how long it will take to reach the destination or how far to travel each day. We would use the distance, rate, and time formula, D = r t , which we have already seen.

In this section, we will use this formula in situations that require a little more algebra to solve than the ones we saw earlier. Generally, we will be looking at comparing two scenarios, such as two vehicles travelling at different rates or in opposite directions. When the speed of each vehicle is constant, we call applications like this uniform motion problems .

Our problem-solving strategies will still apply here, but we will add to the first step. The first step will include drawing a diagram that shows what is happening in the example. Drawing the diagram helps us understand what is happening so that we will write an appropriate equation. Then we will make a table to organize the information, like we did for the money applications.

The steps are listed here for easy reference:

Use a problem-solving strategy in distance, rate, and time applications.

  1. Read the problem. Make sure all the words and ideas are understood.
    • Draw a diagram to illustrate what it happening.
    • Create a table to organize the information.
    • Label the columns rate, time, distance.
    • List the two scenarios.
    • Write in the information you know.
    A table with three rows and four columns and an extra cell at the bottom of the fourth column. The first row is a header row and reads from left to right _____, Rate, Time, and Distance. The rest of the cells are blank.
  2. Identify what we are looking for.
  3. Name what we are looking for. Choose a variable to represent that quantity.
    • Complete the chart.
    • Use variable expressions to represent that quantity in each row.
    • Multiply the rate times the time to get the distance.
  4. Translate into an equation.
    • Restate the problem in one sentence with all the important information.
    • Then, translate the sentence into an equation.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in 4 hours and the local train takes 5 hours for the trip. The speed of the express train is 12 miles per hour faster than the speed of the local train. Find the speed of both trains.

Solution

Step 1. Read the problem. Make sure all the words and ideas are understood.

  • Draw a diagram to illustrate what it happening. Shown below is a sketch of what is happening in the example.

    Pittsburgh and Washington, DC, are represented by two separate lines. There is a line marked Express Train from Pittsburgh to Washington that is 12 mph faster and 4 hours long. There is a line marked Local Train from Pittsburgh to Washington that take 5 hours. The space between Pittsburgh and Washington is marked distance.
    A table with three rows and four columns. The first row is a header row and reads from left to right _____, Rate (mph), Time (hrs), and Distance (miles). Below the blank header cell, we have Express and then Local. Below the Time header cell, we have 4 and then 5. The rest of the cells are blank.
  • Create a table to organize the information.
  • Label the columns “Rate,” “Time,” and “Distance.”
  • List the two scenarios.
  • Write in the information you know.

Step 2. Identify what we are looking for.

  • We are asked to find the speed of both trains.
  • Notice that the distance formula uses the word “rate,” but it is more common to use “speed” when we talk about vehicles in everyday English.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

  • Complete the chart
  • Use variable expressions to represent that quantity in each row.
  • We are looking for the speed of the trains. Let’s let r represent the speed of the local train. Since the speed of the express train is 12 mph faster, we represent that as r + 12 .

r = speed of the local train r + 12 = speed of the express train

Fill in the speeds into the chart.

A table with three rows and four columns. The first row is a header row and reads from left to right _____, Rate (mph), Time (hrs), and Distance (miles). Below the blank header cell, we have Express and then Local. Below the Rate header cell, we have r plus 12 and then r. Below the Time header cell, we have 4 and then 5. The rest of the cells are blank.

Multiply the rate times the time to get the distance.

A table with three rows and four columns. The first row is a header row and reads from left to right _____, Rate (mph), Time (hrs), and Distance (miles). Below the blank header cell, we have Express and then Local. Below the Rate header cell, we have r plus 12 and then r. Below the Time header cell, we have 4 and then 5. Below the Distance header cell, we have 4 times the quantity (r plus 12) and then 5r.

Step 4. Translate into an equation.

  • Restate the problem in one sentence with all the important information.
  • Then, translate the sentence into an equation.
  • The equation to model this situation will come from the relation between the distances. Look at the diagram we drew above. How is the distance travelled by the express train related to the distance travelled by the local train?
  • Since both trains leave from Pittsburgh and travel to Washington, D.C. they travel the same distance. So we write:


The sentence, “The distance traveled by the express train equals the distance traveled by the local train,” can be translated to an equation. Translate “distance traveled by the express train” to 4 times the quantity r plus 12, and translate “distance traveled by the local train” to 5r. The full equation is 4 times the quantity r plus 12 equals 5r.

Step 5. Solve the equation using good algebra techniques.

Now solve this equation. .
.
.
So the speed of the local train is 48 mph.
Find the speed of the express train. .
.
.
The speed of the express train is 60 mph.

Step 6. Check the answer in the problem and make sure it makes sense.

express train 60 mph (4 hours) = 240 miles local train 48 mph (5 hours) = 240 miles ✓

Step 7. Answer the question with a complete sentence.

  • The speed of the local train is 48 mph and the speed of the express train is 60 mph.
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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