It is proposed to introduce daily trains between Chennai and New Delhi in both directions simultaneously, departing at each station at 6.00PM. If the travel time is 36 hours, what is the minimum number of rakes (train sets) required to run the service?
4
Rakes Required for Daily Train Service
The question asks for the minimum number of rakes (train sets) required to run a daily train service simultaneously in both directions between Chennai and New Delhi, given the departure schedule and travel time.
Here's how we can determine the minimum number of rakes:
- Understand the Service Schedule: Trains depart daily at 6:00 PM from both Chennai and New Delhi. This means a train leaves Chennai every 24 hours and a train leaves New Delhi every 24 hours. The service frequency is 24 hours.
- Understand the Travel Time: The travel time for a one-way journey is 36 hours.
- Calculate the Rake Cycle Time: Consider one rake starting from Chennai.
- It travels from Chennai to New Delhi: 36 hours.
- Upon reaching New Delhi, it needs to be ready for the return journey. Since trains depart New Delhi at 6:00 PM daily, and a rake arriving at 6:00 AM (after a 36-hour journey from Chennai starting at 6:00 PM two days prior) can potentially be used for the 6:00 PM departure on the same day, the minimum effective turnaround time at New Delhi is from 6:00 AM to 6:00 PM, which is 12 hours.
- It travels from New Delhi back to Chennai: 36 hours.
- Upon reaching Chennai, it needs to be ready for its next departure from Chennai. Similar to New Delhi, a rake arriving at 6:00 AM (after a 36-hour journey from New Delhi starting at 6:00 PM two days prior) can potentially be used for the 6:00 PM departure on the same day. The minimum effective turnaround time at Chennai is also 12 hours.
The total cycle time for one rake, from leaving Chennai to being ready for its next departure from Chennai, is the sum of travel times and turnaround times:
Total Cycle Time = Travel time (C to D) + Turnaround at D + Travel time (D to C) + Turnaround at C
Total Cycle Time = $36 \text{ hours} + 12 \text{ hours} + 36 \text{ hours} + 12 \text{ hours} = 96 \text{ hours}$
Minimum Number of Rakes Calculation
To maintain a daily service (frequency of 24 hours), the number of rakes required is the total cycle time divided by the service frequency.
Minimum Rakes Required = $\frac{\text{Total Cycle Time}}{\text{Service Frequency}}$
Minimum Rakes Required = $\frac{96 \text{ hours}}{24 \text{ hours}}$
Minimum Rakes Required = $4$
This calculation shows that 4 rakes are needed to ensure a rake is available at Chennai every 24 hours for the 6:00 PM departure, considering the travel time and return journey. Since the service is simultaneous and symmetrical from New Delhi, the same number of rakes supports the New Delhi departures as well.
Visualizing Rake Movement
Let's track the position of 4 rakes over a few days, assuming Rakes 1, 2, 3, 4 are used sequentially for Chennai departures starting Day 1:
| Day | Time | Event | Rake Status |
|---|---|---|---|
| Day 1 | 6:00 PM | Train leaves Chennai (C) | Rake 1: C $\to$ D (en route) Rake 2: Available Rake 3: Available Rake 4: Available |
| Day 2 | 6:00 PM | Train leaves Chennai (C) | Rake 1: C $\to$ D (en route) Rake 2: C $\to$ D (en route) Rake 3: Available Rake 4: Available |
| Day 3 | 6:00 AM | Train arrives New Delhi (D) | Rake 1: At D (available for return) |
| Day 3 | 6:00 PM | Train leaves Chennai (C) Train leaves New Delhi (D) |
Rake 1: D $\to$ C (en route) Rake 2: C $\to$ D (en route) Rake 3: C $\to$ D (en route) Rake 4: Available |
| Day 4 | 6:00 AM | Train arrives New Delhi (D) | Rake 2: At D (available for return) |
| Day 4 | 6:00 PM | Train leaves Chennai (C) Train leaves New Delhi (D) |
Rake 1: D $\to$ C (en route) Rake 2: D $\to$ C (en route) Rake 3: C $\to$ D (en route) Rake 4: C $\to$ D (en route) |
| Day 5 | 6:00 AM | Train arrives Chennai (C) Train arrives New Delhi (D) |
Rake 1: At C (available for next C departure) Rake 3: At D (available for next D departure) |
| Day 5 | 6:00 PM | Train leaves Chennai (C) Train leaves New Delhi (D) |
Rake 1: C $\to$ D (en route) Rake 2: D $\to$ C (en route) Rake 3: D $\to$ C (en route) Rake 4: C $\to$ D (en route) |
At any point in time after the service is fully running, you need:
- 1 rake en route from Chennai to New Delhi (e.g., Rake 4 on Day 4 PM).
- 1 rake en route from New Delhi to Chennai (e.g., Rake 1 on Day 4 PM).
- 1 rake at Chennai, ready for the next departure (e.g., Rake 1 available at C on Day 5 AM, used Day 5 PM).
- 1 rake at New Delhi, ready for the next departure (e.g., Rake 3 available at D on Day 5 AM, used Day 5 PM).
This confirms that a minimum of 4 rakes are required.
Important Questions from Relative Speed
The driver of a car, which is travelling at a speed of 75 km/h, locates a bus 80 m ahead of him, travelling in the same direction. After 15 seconds, he finds that the bus is 40 m behind the car. What is the speed of the bus (in km/h)?
The distance between two station A and B is 800 km. A train X starts from A and moves towards B at 40 km/h and another trains Y starts from B and moves towards A at 60 km/h. how far from A will they cross each other?
A and B are travelling towards each other from the points P and Q respectively. After crossing each other, A and B take \(6\frac{1}{8}\) hours and 8 hours, respectively, to reach their destinations Q and P, respectively. If the speed of B is 16.8 km/h, then the speed (in km/hr) of A is:
A train leaves station A at 8 am and reaches station B at 12 noon. A car leaves station B at 8:30 am and reaches station A at the same time when the train reaches station B. At what time do they meet?
A and B start moving towards each other from places X and Y respectively, at the same time. The speed of A is 20% more than that of B. After meeting on the way, A and B take \(2\frac{1}{2}\) hours and x hours now to reach Y and X respectively. What is the value of x?
