If a train crosses a km-stone in 12 seconds, how long will it take to cross 91 km stone completely if its speed is 60 km/h?
1 h 30 min 12 sec
Solving the Train Speed and Distance Problem
This problem involves a train's speed, its length, and the time it takes to cover a certain distance marked by km-stones. We are given the time it takes to cross a single km-stone and the train's speed, and we need to find the total time to cross 91 km-stones.
Understanding 'Crossing a Km-Stone'
A km-stone can be considered a point. When a train crosses a point, the time taken is equal to the time required for the entire length of the train to pass that point. This means the distance covered during this time is equal to the train's length.
Calculating the Train's Length
We are given that the train crosses a km-stone in 12 seconds and its speed is 60 km/h.
- Speed = 60 km/h
- Time to cross 1 km-stone = 12 seconds
First, let's convert the speed from km/h to meters per second (m/s), as the time is given in seconds.
\(\text{Speed} = 60 \, \text{km/h} = 60 \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{h}}{3600 \, \text{s}}\)
\(\text{Speed} = \frac{60 \times 1000}{3600} \, \text{m/s} = \frac{60000}{3600} \, \text{m/s} = \frac{600}{36} \, \text{m/s} = \frac{100}{6} \, \text{m/s} = \frac{50}{3} \, \text{m/s}\)
The distance covered when crossing a point is the train's length (L). Using the formula Distance = Speed \(\times\) Time:
\(L = \text{Speed} \times \text{Time to cross point}\)
\(L = \frac{50}{3} \, \text{m/s} \times 12 \, \text{s}\)
\(L = 50 \times \frac{12}{3} \, \text{m} = 50 \times 4 \, \text{m} = 200 \, \text{m}\)
So, the length of the train is 200 meters.
Calculating Time to Cross 91 Km-Stones
We need to find the time it takes the train to cross 91 km-stones completely. Let's assume the train starts its journey just as it crosses the first km-stone. To reach and completely cross the 91st km-stone, the train's front must travel the distance from the 1st km-stone to the 91st km-stone, and then the entire train's length must pass the 91st km-stone.
Assuming km-stones are placed 1 km apart, the distance between the 1st km-stone and the 91st km-stone is:
Distance between stones = \((91 - 1) \, \text{km} = 90 \, \text{km}\)
This 90 km is the distance the front of the train travels to reach the 91st km-stone.
We need to calculate the time taken to travel this 90 km distance at a speed of 60 km/h.
\(\text{Time to travel 90 km} = \frac{\text{Distance}}{\text{Speed}}\)
\(\text{Time to travel 90 km} = \frac{90 \, \text{km}}{60 \, \text{km/h}} = \frac{90}{60} \, \text{h} = \frac{3}{2} \, \text{h} = 1.5 \, \text{h}\)
\(1.5 \, \text{h} = 1 \, \text{hour} + 0.5 \, \text{hours}\)
\(0.5 \, \text{hours} = 0.5 \times 60 \, \text{minutes} = 30 \, \text{minutes}\)
So, the time taken to travel 90 km is 1 hour and 30 minutes. This is the time until the front of the train reaches the 91st km-stone.
To 'cross 91 km stone completely', the entire train of length 200m must pass the 91st km-stone. We already know that the time taken for the train to cross a single km-stone (which is equivalent to its length passing a point) is 12 seconds.
Total time to cross 91 km-stones = (Time to travel from 1st to 91st km-stone) + (Time to cross the 91st km-stone)
Total time = \(1 \, \text{hour} \, 30 \, \text{minutes} + 12 \, \text{seconds}\)
Total time = 1 hour 30 minutes 12 seconds.
Let's summarise the steps:
- Convert speed to m/s.
- Use the time to cross one km-stone (12s) and speed to find the train's length.
- Calculate the distance between the 1st and 91st km-stones (90 km).
- Calculate the time taken to travel this 90 km distance.
- Add the time taken for the train's length to clear the 91st km-stone (which is the same as crossing one km-stone).
Calculations:
Speed = 60 km/h = \(50/3\) m/s
Train Length (L) = Speed \(\times\) Time = \( \frac{50}{3} \, \text{m/s} \times 12 \, \text{s} = 200 \, \text{m}\)
Distance between 1st and 91st km-stones = 90 km
Time to travel 90 km = \( \frac{90 \, \text{km}}{60 \, \text{km/h}} = 1.5 \, \text{h} = 1 \, \text{h} \, 30 \, \text{min}\)
Time to cross 91st km-stone completely = 12 seconds (given)
Total Time = Time to travel 90 km + Time to cross 91st stone
Total Time = 1 h 30 min + 12 sec = 1 h 30 min 12 sec.
Comparing this with the given options, we find that 1 hour 30 minutes 12 seconds is one of the options.
| Metric | Value | Units |
|---|---|---|
| Train Speed | 60 | km/h |
| Train Speed | \(50/3\) | m/s |
| Time to cross 1 km-stone | 12 | seconds |
| Train Length | 200 | meters |
| Distance between 1st and 91st km-stone | 90 | km |
| Time to travel 90 km | 1 h 30 min | h/min |
| Total Time | 1 h 30 min 12 sec | h/min/sec |
Revision Table: Train Speed Calculation
| Concept | Formula | Application |
|---|---|---|
| Speed Conversion (km/h to m/s) | Speed (m/s) = Speed (km/h) \(\times \frac{1000}{3600}\) | Convert 60 km/h to m/s |
| Distance, Speed, Time | Distance = Speed \(\times\) Time | Calculate train length or time taken |
| Crossing a point | Time = Train Length / Speed | Used to find train length from crossing time |
| Crossing a long object/distance | Total Distance = Distance of object/path + Train Length | Used for bridges, platforms, distance between points |
Additional Information: Train Problems Concepts
Train problems in quantitative aptitude usually involve the concepts of speed, time, distance, and relative speed, often considering the length of the train itself. Here are some key points:
- Crossing a point object (like a pole, a man, a km-stone): The distance covered by the train is equal to its own length. Time = Train Length / Speed.
- Crossing a length (like a bridge, platform, tunnel, or another train): The total distance covered is the sum of the train's length and the length of the object it is crossing. Time = (Train Length + Object Length) / Speed.
- Relative Speed (Trains moving in the same direction): If two trains are moving in the same direction with speeds \(S_1\) and \(S_2\) (\(S_1 > S_2\)), their relative speed is \(S_1 - S_2\). When one train crosses another, the distance is the sum of their lengths.
- Relative Speed (Trains moving in opposite directions): If two trains are moving in opposite directions with speeds \(S_1\) and \(S_2\), their relative speed is \(S_1 + S_2\). When they cross each other, the distance is the sum of their lengths.
In this specific problem involving km-stones, crossing the distance between two stones is a standard distance/speed/time calculation, but 'crossing' the stone itself involves the train's length.
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