Uniform Motion: Equal times
Uniform motion problems where two travelers spend equal amounts of time set up the equation d1/r1 = d2/r2, since t = d/r and both times are the same. A typical scenario involves two vehicles that leave simultaneously and travel until they have covered distances that satisfy a given condition. Chapter 8 of OpenStax Elementary Algebra 2E presents equal-time motion problems as applications of rational equations. Recognizing which setup applies — equal time or sum of times — is the key decision that determines the entire equation structure.
Key Concepts
Property When solving uniform motion problems where the time is the same for two different trips, we use the formula $t = \frac{D}{r}$ for each trip and set them equal. Let $t 1$ and $t 2$ be the times for two trips. If $t 1 = t 2$, then: $$ \frac{D 1}{r 1} = \frac{D 2}{r 2} $$ This is common in problems involving a headwind (speed is $r w$) and a tailwind (speed is $r+w$), where $r$ is the vehicle's speed and $w$ is the wind or current speed.
Examples An airplane flies 480 miles with a 40 mph tailwind in the same time it flies 320 miles against it. Let $r$ be the plane's speed. The equation is $\frac{480}{r+40} = \frac{320}{r 40}$. Solving gives $160r = 32000$, so $r=200$ mph.
A boat travels 45 miles downstream with a 4 mph current in the same time it travels 27 miles upstream. Let $b$ be the boat's speed. The equation is $\frac{45}{b+4} = \frac{27}{b 4}$. Solving gives $18b = 288$, so $b=16$ mph.
Common Questions
What is a uniform motion equal-times problem?
It is a problem where two objects travel for exactly the same duration. Because time is equal, you set t1 = t2, which means d1/r1 = d2/r2 after substituting t = d/r.
How do I set up an equal-time motion equation?
Write t = d/r for each object. Set the two time expressions equal: d1/r1 = d2/r2. Then cross-multiply or use the LCD to solve for the unknown.
What is the difference between equal-time and sum-of-times motion problems?
In equal-time problems, t1 = t2. In sum-of-times problems, t1 + t2 = total. The word problem context determines which setup to use.
How do I solve d1/r1 = d2/r2?
Cross-multiply to get d1 times r2 = d2 times r1, then solve for the unknown variable. Always check the solution makes physical sense.
When do students learn equal-time motion problems?
These are algebra 1 topics covered in OpenStax Elementary Algebra 2E Chapter 8: Rational Expressions and Equations.
What is a common mistake in equal-time motion problems?
Setting up time as rate times distance instead of distance divided by rate. The formula is always t = d/r.
Can equal-time motion problems involve different directions?
Yes. Two objects traveling in opposite directions for equal time form meeting-point problems, where total distance equals d1 plus d2.