Solving Motion Problems Involving Two Rates
A motion problem is one in which an object is moving at a specific rate for a specific period of time. Motion problems are often solved by using the distance formula:
In motion problems that involve two rates, such as two trains traveling different speeds, let the variable represent one of the unknown quantities and then represent the second quantity in terms of the first.
Example
Justin and Maryanne both rent kayaks to paddle down the river. Both kayaks start at the same time from the same point and travel in the same direction. Justin paddles his kayak at 2 miles per hour, and Maryanne paddles her kayak at 4 miles per hour. In how many hours will the two kayaks be 5 miles apart?
Understand
The question asks to find the time it takes for the kayaks to be separated by 5 miles.
Let t = the time when kayaks are 5 miles apart.
The chart below establishes the rate, time, and distance for each kayak.
Since the kayaks are traveling in the same direction, the distance between them is found by subtracting the distance traveled by the slower kayak from the distance traveled by the faster kayak.
Translate
(distance traveled by faster kayak) − (distance traveled by slower kayak) = 5 miles
4t − 2t = 5 miles
Solve
4t − 2t = 5
2t = 5
t = 2.5
Answer
After 2.5 hours the two kayaks will be 5 miles apart.
Practice
1) Two trains in Toronto start at the same station going in the same direction on sets of parallel tracks. The local train stops and averages 30 km per hour. The express train stops less frequently and averages 50 km per hour. In how many hours will the two trains be 6 km apart?
2) Two friends go horseback riding on the same trail in the same direction. Kelly's horse travels at 5 km per hour while Karen's horse travels at a slower pace. After 2 hours they are 3 km apart. Find the speed at which Karen's horse is traveling.
3) Rich and Maggie Roberts have walkie-talkies that have a range of 16.8 miles. Rich and Maggie start at the same point and walk in opposite directions. If Rich walks 3 mph and Maggie walks 4 mph, how long will it take before they are out of range?
4) Two highway paving crews are 20 miles apart working toward each other. One crew paves 0.4 miles of road per day more than the other crew. The two crews meet after 10 days. Find the rate at which each crew paves the road.
Answers
1. 0.3 h
2. 3.5 km/h
3. 2.4 h
4. 0.8 miles/day and 1.2 miles/day
