3.5 Solve Uniform Motion Applications

Solving Uniform Motion Applications

Distance, Rate, and Time Formula

The core relationship behind every uniform motion problem is one formula:

$d = rt$

This says that distance ($d$) equals rate ($r$) multiplied by time ($t$). "Uniform motion" just means the speed stays constant throughout the trip.

You can rearrange this formula depending on what you're solving for:

• Solve for rate: $r = \frac{d}{t}$
• Solve for time: $t = \frac{d}{r}$

Before plugging in numbers, make sure your units match. If distance is in miles and time is in hours, your rate will be in miles per hour. If someone gives you time in minutes but distance in miles, convert minutes to hours first (divide by 60).

Example: A car drives 180 miles in 3 hours. What's the rate?

$r = \frac{180}{3} = 60 \text{ mph}$

Using Tables to Organize Motion Problems

Most uniform motion word problems involve two or more travelers, and the information can get tangled fast. A table keeps everything straight.

Steps to set up a motion table:

1. Create columns for Distance, Rate, and Time.
2. Create one row for each object or traveler in the problem.
3. Fill in whatever values the problem gives you.
4. Use variables for the unknowns, and write expressions using $d = rt$ to connect them.
5. Use the relationship the problem describes (same distance, combined distance, same time, etc.) to write an equation and solve.

Here's a sample table with a mix of known and unknown values:

For the marathon row, solve for rate: $r = \frac{26.2}{4.5} \approx 5.82$ mph.

For the flight row, solve for distance: $d = 500 \times 3.5 = 1{,}750$ miles.

A simple sketch can also help. Draw a line segment representing the path, label the start and end points, and mark the distance, rate, and time along it. This is especially useful when two objects are traveling toward each other or in the same direction.

Speed Comparisons in Uniform Motion

When two objects travel the same distance, you can compare their speeds directly:

1. Calculate each object's rate using $r = \frac{d}{t}$.
2. The object with the higher rate is faster.

Example: Cyclist A rides 30 miles in 2 hours. Cyclist B rides 30 miles in 1.5 hours.

$r_A = \frac{30}{2} = 15 \text{ mph}$

$r_B = \frac{30}{1.5} = 20 \text{ mph}$

Cyclist B is faster. Notice the pattern: for a fixed distance, speed and time are inversely related. Shorter time means higher speed. This comes up in problems comparing transportation methods or figuring out which route is quicker.

Common Types of Uniform Motion Word Problems

Most textbook problems fall into a few categories. Recognizing the type helps you set up the equation.

• Opposite directions: Two objects leave the same point heading away from each other. Their distances add up to the total distance between them. The equation looks like $r_1 t + r_2 t = d_{\text{total}}$.
• Same direction: A faster object chases a slower one (or one leaves earlier). When the faster one catches up, they've covered the same distance. Set $d_1 = d_2$.
• Round trip: An object goes out at one speed and returns at another. The distance out equals the distance back, but the times are different.

For each type, the table method works the same way. Fill in what you know, use $d = rt$ to write expressions for the unknowns, then set up one equation based on how the distances or times relate.

Quick example (opposite directions): Two cars leave the same city at the same time, heading in opposite directions. One drives 55 mph and the other drives 65 mph. How long until they're 360 miles apart?

$55t + 65t = 360$ $120t = 360$ $t = 3 \text{ hours}$