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Difficulty: MediumRelative Motion and Catch-Up Scenarios

A passenger train leaves West Station heading east toward East Station, which is 270270 miles away, traveling at a constant speed of 6060 mph. Exactly 11 hour later, a freight train leaves East Station heading west toward West Station on a parallel track at a constant speed of 4545 mph. How many hours after the passenger train's departure will the two trains pass each other?

Answer: 3 hours

Answer

3 hours
The passenger train travels 60 miles in its first hour, leaving 210 miles between the trains. Traveling toward each other, their combined relative speed is 105 mph. It takes them 2 hours to cover the 210 miles together. Adding the initial 1 hour, the total time since the passenger train departed is 3 hours.

Step-by-Step Solution

1
Calculate the distance covered by the passenger train during its 1-hour head start.
60 mph×1 hour=60 miles60 \text{ mph} \times 1 \text{ hour} = 60 \text{ miles}
This establishes how much distance is covered before the second train begins moving.
2
Find the remaining distance between the two trains.
270 miles−60 miles=210 miles270 \text{ miles} - 60 \text{ miles} = 210 \text{ miles}
This is the remaining gap the two trains must close together once both are moving.
3
Calculate the combined (relative) speed of the two trains.
60 mph+45 mph=105 mph60 \text{ mph} + 45 \text{ mph} = 105 \text{ mph}
Because they are traveling toward each other from opposite directions, their speeds add up to close the distance.
4
Determine the time it takes to close the remaining 210 miles.
210 miles÷105 mph=2 hours210 \text{ miles} \div 105 \text{ mph} = 2 \text{ hours}
Using the formula Time = Distance / Rate.
5
Calculate the total time since the passenger train's departure.
1 hour+2 hours=3 hours1 \text{ hour} + 2 \text{ hours} = 3 \text{ hours}
The question specifically asks for the time after the passenger train's departure, which includes its initial 1-hour head start.

Key Concept

Relative motion and converging objects with staggered start times
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Topics

Relative Motion and Catch-Up Scenarios

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