8.8 Solve Uniform Motion and Work Applications

Uniform Motion

Uniform Motion Formula Applications

Uniform motion problems all revolve around one core relationship: distance equals rate times time.

$d = rt$

You can rearrange this formula depending on which variable you need to find:

• Distance unknown: $d = rt$
• Rate unknown: $r = \frac{d}{t}$
• Time unknown: $t = \frac{d}{r}$

Here's how to work through a uniform motion problem:

1. Read the problem and identify what you know (two of the three variables) and what you need to find.
2. Check that your units are consistent. If speed is in miles per hour but time is given in minutes, convert minutes to hours first.
3. Plug the known values into the appropriate version of the formula.
4. Solve for the unknown variable.
5. State your answer with units and check that it makes sense in context.

Example: A car travels 150 miles at 50 mph. How long does the trip take?

$t = \frac{d}{r} = \frac{150}{50} = 3 \text{ hours}$

A common source of errors is mismatched units. If a problem gives you a rate in feet per second and a distance in miles, you need to convert one or the other before plugging into the formula.

Setting Up Rational Equations for Motion

Many problems in this section involve two travelers, or the same traveler going and returning at different speeds. These problems typically lead to rational equations because you're expressing time as $t = \frac{d}{r}$, which creates fractions.

A typical setup: Two cyclists leave the same point traveling in opposite directions. One rides at 12 mph and the other at 8 mph. How long until they are 60 miles apart?

Since they travel in opposite directions, their distances add up:

$12t + 8t = 60$ $20t = 60$ $t = 3 \text{ hours}$

For problems where a current or wind affects speed, the key idea is that the effective rate changes:

• Traveling with a current of speed $c$: effective rate = $r + c$
• Traveling against a current of speed $c$: effective rate = $r - c$

If a boat's still-water speed is $r$ and it travels 24 miles upstream and 24 miles downstream in the same total time, you'd set up an equation using $t = \frac{d}{r}$ for each leg of the trip.

Work Applications

Collaborative Work Time Calculations

Work problems use the idea of a work rate, which is the fraction of a job someone completes per unit of time. If a painter can finish a room in 6 hours, that painter's work rate is $\frac{1}{6}$ of the job per hour.

When people work together, you add their individual rates to get the combined rate:

$\text{Combined rate} = \frac{1}{t_1} + \frac{1}{t_2}$

where $t_1$ and $t_2$ are the times each person would take working alone.

To find how long the job takes together, set the combined work equal to 1 (one complete job):

$\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{t}$

where $t$ is the time working together.

Step-by-step process:

1. Find each worker's individual rate. If Person A finishes in 4 hours, their rate is $\frac{1}{4}$. If Person B finishes in 6 hours, their rate is $\frac{1}{6}$.
2. Add the rates: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$
3. Set the combined rate equal to $\frac{1}{t}$: $\frac{5}{12} = \frac{1}{t}$
4. Solve: $t = \frac{12}{5} = 2.4$ hours (or 2 hours 24 minutes).
5. Check: the answer should be less than the faster worker's time alone (4 hours). It is, so this makes sense.

That final check is worth remembering. Working together should always be faster than either person working alone. If your answer is longer, something went wrong.

Multi-Step Rate Problems

Some problems combine motion and work concepts, or involve multiple stages. The strategy is to break the problem into pieces.

1. Identify each stage of the problem separately. What information applies to each stage?

2. Write an equation for each stage using the appropriate formula ($d = rt$ for motion, work rate formulas for jobs).

3. Look for a relationship that connects the stages. Common connections include:

   • Total time for all stages equals a given amount
   • Total distance equals a given amount
   • Two times are equal (e.g., "the trip there took the same time as the trip back")

4. Use that relationship to write one equation in one variable, then solve.

5. Answer the original question and verify that your solution is reasonable.

Example: A plumber and an apprentice are fixing pipes. The plumber works alone for 2 hours, then the apprentice joins. The plumber can do the whole job in 5 hours; the apprentice can do it in 8 hours. How long after the apprentice joins do they finish?

• In the first 2 hours, the plumber completes $\frac{2}{5}$ of the job.
• Remaining work: $1 - \frac{2}{5} = \frac{3}{5}$
• Together, their rate is $\frac{1}{5} + \frac{1}{8} = \frac{13}{40}$ per hour.
• Time to finish: $\frac{3}{5} \div \frac{13}{40} = \frac{3}{5} \times \frac{40}{13} = \frac{120}{65} = \frac{24}{13} \approx 1.85$ hours.

Breaking the problem into stages like this keeps things organized and prevents you from losing track of what each number represents.