3.5 Solve uniform motion applications

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.
  
  
  
  
• 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 $r+12.$

$\begin{array}{ccc}\hfill r& =\hfill & \text{speed of the local train}\hfill \\ \hfill r+12& =\hfill & \text{speed of the express train}\hfill \end{array}$

Fill in the speeds into the chart.

Multiply the rate times the time to get the distance.

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:

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.

$\begin{array}{cccc}\text{express train}\hfill & & & \text{60 mph (4 hours) = 240 miles}\hfill \\ \text{local train}\hfill & & & \text{48 mph (5 hours) = 240 miles ✓}\hfill \end{array}$

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.