Lesson 3.5: Solve Uniform Motion Applications

New Concept

Learn to solve uniform motion problems using the formula $D = rt$. We'll analyze scenarios with two moving objects, organize data in tables, and create equations based on their distances to find unknown speeds, times, or distances.

What’s next

Next, you'll tackle interactive examples and practice problems, using tables and diagrams to set up and solve equations for various uniform motion scenarios.

Property

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, $D = rt$. When the speed of each vehicle is constant, we call applications like this uniform motion problems.

Problem-Solving Strategy:

1. Read and Understand: Draw a diagram and create a table with columns for Rate, Time, and Distance for each scenario.
2. Identify Goal: Determine what you need to find.
3. Name and Assign Variables: Represent unknown quantities with variables and complete the table.
4. Translate to an Equation: Relate the distances based on the problem (e.g., they are equal, or their sum is a total).
5. Solve: Use algebra to solve the equation.
6. Check: Ensure the answer is reasonable.
7. Answer: State the answer in a complete sentence.

Examples

• Two trains travel the same 300-mile route. The express train is 20 mph faster and arrives 1 hour sooner. Set up an equation based on their travel times to find their speeds.
• A car and a truck leave the same city, traveling in opposite directions. The car travels at 65 mph and the truck at 55 mph. Their combined distance is $D_{car} + D_{truck}$. To find when they are 360 miles apart, solve $65t + 55t = 360$.
• A cyclist rides uphill at 8 mph and downhill at 16 mph. If the trip to the top of the hill and back covers the same path, the distance is equal: $8t_{up} = 16t_{down}$.

Property

In some uniform motion problems, the situation involves two objects or trips that cover the same distance. Because the distances are equal, we can set them equal to each other to form an equation.

Formula:

Examples

• A train takes 4 hours to complete a journey, while a car takes 5 hours. The train is 15 mph faster than the car. Since the distance is the same, $4(r+15) = 5r$. The car's speed is 60 mph and the train's is 75 mph.
• Jenna walks to the library in 40 minutes. If she bikes, it takes her 15 minutes. Her biking speed is 5 mph faster than her walking speed. The distances are equal: $\frac{2}{3}r = \frac{1}{4}(r+5)$. Her walking speed is 3 mph.
• A boat travels upstream against a current and then returns downstream with the current. The distance traveled each way is identical. If the boat's speed is $b$ and the current's is $c$, then $D_{up} = (b-c)t_{up}$ and $D_{down} = (b+c)t_{down}$.

Property

When two objects start at the same point and travel in opposite directions, the distance between them is the sum of the distances each object has traveled.

Formula:

Examples

• Two cars leave a gas station at the same time, one heading east at 60 mph and the other west at 70 mph. To find when they are 325 miles apart, solve $60t + 70t = 325$. It takes 2.5 hours.
• Cindy rides her bike north at 15 mph, and Richard rides south at 12 mph. They start from their dorm at the same time. The time it takes for them to be 54 miles apart is found by solving $15t + 12t = 54$, which gives $t = 2$ hours.
• Two jets leave from the same airport. One flies north at 500 mph and the other flies south at 550 mph. After 3 hours, the distance between them is $(500)(3) + (550)(3) = 1500 + 1650 = 3150$ miles.

Property

When two objects start at different points and travel towards each other, the sum of their distances traveled equals the initial distance between them.

Formula:

Examples

• Two friends live 150 miles apart and drive towards each other to meet. Alex drives at 65 mph for 1.5 hours, and Ben drives at 55 mph for 1 hour. The total distance covered is $1.5(r+10) + 1r = 150$, solving for Ben's speed $r$.
• A trip of 300 miles is split into two parts. The first part is in city traffic for 2 hours, and the second is on the highway for 3 hours. If the highway speed is 30 mph faster than the city speed, we solve $2r + 3(r+30) = 300$ to find the speeds.
• Saul and Erwin drive towards each other from cities 500 miles apart. Saul drives for 3 hours, and Erwin for 4 hours. Erwin is 10 mph slower than Saul. The equation is $3r + 4(r-10) = 500$. Saul's speed is 77.1 mph.

Property

It is important to make sure the units match when we use the distance rate and time formula. For instance, if the rate is in miles per hour, then the time must be in hours.

Examples

• To use a time of 45 minutes with a speed in mph, you must convert the time to hours: $45 \text{ minutes} = \frac{45}{60} \text{ hours} = 0.75 \text{ hours}$.
• A car travels at 50 mph for 90 minutes. Before calculating distance, convert the time to $1.5$ hours. The distance is $D = 50 \times 1.5 = 75$ miles.
• A jogger runs for 15 minutes at a speed of 8 miles per hour. To find the distance, convert 15 minutes to $\frac{15}{60} = \frac{1}{4}$ hour. The distance is $D = 8 \times \frac{1}{4} = 2$ miles.

Explanation

If your speed is in miles per hour, your time must be in hours, not minutes. Always convert units before calculating! For example, 30 minutes becomes $0.5$ hours. Mismatched units will give you the wrong answer every time.