When two bodies are moving in opposite directions, whether towards each other or away from each other, the rate at which the distance between them is changing, whether growing larger or smaller, is the sum of their individual rates.
Examples
Two cars meet at an intersection, one heading north; the other, south. If the northbound driver drives at 30 mph and the southbound driver at 40 mph, when will they be 35 miles apart?
The distance between them is growing at the rate of 30 + 40 = 70 mph. The question then becomes, ‘‘how long will it take a body moving 70 mph to travel 35 miles?’’
Let t represent the number of hours the cars travel after leaving the intersection.
In half an hour, the cars will be 35 miles apart.
Katy left her house on a bicycle heading north at 8 mph. At the same time, her sister Molly headed south at 12 mph. How long will it take for them to be 24 miles apart?
The distance between them is increasing at the rate of 8 + 12 = 20 mph. The question then becomes ‘‘How long will it take a body moving 20 mph to travel 24 miles?’’
Let t represent the number of hours each girl is traveling.
The girls will be 24 miles apart after 1 ⅕ hours or 1 hour 12 minutes.
Practice
1. Two airplanes leave an airport simultaneously, one heading east; the other, west. The eastbound plane travels at 140 mph and the westbound plane travels at 160 mph. How long will it take for the planes to be 750 miles apart?
2. Mary began walking home from school, heading south at a rate of 4 mph. Sharon left school at the same time heading north at 6 mph. How long will it take for them to be 3 miles apart?
3. Two freight trains pass each other on parallel tracks. One train is traveling west, going 40 mph. The other is traveling east, going 60 mph. When will the trains be 325 miles apart?
Solutions
1. The planes are moving apart at a rate of 140 + 160 = 300 mph. Let t represent the number of hours the planes are flying.
In 2 ½ hours, or 2 hours 30 minutes, the planes will be 750 miles apart.
2. The distance between the girls is increasing at the rate of 4 + 6 = 10 mph. Let t represent the number of hours the girls are walking.
Mary and Sharon will be 3 miles apart in 3/10 of an hour or 6(3/10) = 18 minutes.
3. The distance between the trains is increasing at the rate of 40 + 60 = 100 mph. Let t represent the number of hours the trains travel after leaving the station.
The trains will be 325 miles apart after 3 ¼ hours or 3 hours 15 minutes.
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