Solving for Train Meeting Distance from Delhi
This problem involves calculating the distance from Delhi where two trains, Train A and Train B, will meet. They depart from Delhi to Hyderabad at different times and travel at different speeds. We need to determine the distance from Delhi at which their paths converge.
Understanding the Train Travel Details
- Train A: Departs Delhi at 7:00 a.m. with a speed of 80 kmph.
- Train B: Departs Delhi at 7:50 a.m. with a speed of 100 kmph.
- Direction: Both trains travel from Delhi towards Hyderabad.
- Goal: Find the distance from Delhi where they meet.
Calculating Train A's Head Start
Train B starts 50 minutes after Train A. Before Train B even begins its journey, Train A has already covered some distance. Let's calculate this 'head start' distance.
The time difference between their departures is 50 minutes.
Convert this time difference into hours for consistency with speed units (kmph):
Time difference = 50 minutes = \(\frac{50}{60}\) hours = \(\frac{5}{6}\) hours.
Now, calculate the distance Train A travels during this time:
Distance = Speed × Time
Head start distance for Train A = \(80 \text{ kmph} \times \frac{5}{6} \text{ hours}\)
Head start distance = \(\frac{400}{6}\) km = \(\frac{200}{3}\) km.
So, at 7:50 a.m., when Train B starts from Delhi, Train A is already \(\frac{200}{3}\) km away from Delhi.
Determining the Relative Speed
Since Train B is faster than Train A and they are moving in the same direction, Train B will eventually catch up. The rate at which Train B closes the distance between them is called the relative speed.
Relative Speed = Speed of the faster train - Speed of the slower train
Relative Speed = Speed of Train B - Speed of Train A
Relative Speed = \(100 \text{ kmph} - 80 \text{ kmph}\)
Relative Speed = \(20 \text{ kmph}\).
This means Train B gains 20 kilometers on Train A every hour.
Calculating the Time to Meet
To find out how long it takes for Train B to catch up, we divide the distance Train B needs to cover (which is Train A's head start distance) by the relative speed.
Time to meet = \(\frac{\text{Head start distance}}{\text{Relative Speed}}\)
Time to meet = \(\frac{\frac{200}{3} \text{ km}}{20 \text{ kmph}}\)
Time to meet = \(\frac{200}{3 \times 20}\) hours
Time to meet = \(\frac{10}{3}\) hours.
This is the duration Train B travels from its starting time (7:50 a.m.) until it meets Train A.
Calculating the Meeting Distance from Delhi
Now, we can find the total distance from Delhi where the trains meet. We can use the information for Train B: its speed and the time it traveled until the meeting point.
Distance from Delhi = Speed of Train B × Time to meet
Distance from Delhi = \(100 \text{ kmph} \times \frac{10}{3} \text{ hours}\)
Distance from Delhi = \(\frac{1000}{3}\) km.
Verification Using Train A's Journey
We can double-check this result using Train A's travel details. Train A travels for the initial 50 minutes (\(\frac{5}{6}\) hours) plus the time it took Train B to catch up (\(\frac{10}{3}\) hours).
Total time Train A travels = \(\frac{5}{6} \text{ hours} + \frac{10}{3} \text{ hours}\)
To add these fractions, find a common denominator (which is 6):
Total time Train A travels = \(\frac{5}{6} + \frac{10 \times 2}{3 \times 2} = \frac{5}{6} + \frac{20}{6} = \frac{25}{6}\) hours.
Now, calculate the distance Train A covers in this total time:
Distance = Speed of Train A × Total time Train A travels
Distance = \(80 \text{ kmph} \times \frac{25}{6} \text{ hours}\)
Distance = \(\frac{80 \times 25}{6}\) km = \(\frac{2000}{6}\) km = \(\frac{1000}{3}\) km.
Both methods give the same result, confirming the accuracy of our calculation.
Conclusion on Train Meeting Distance
The two trains will be together after traveling \(\frac{1000}{3}\) kilometers from Delhi.
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