The time taken by a train to cross a man travelling in another train is 10 seconds, when the other train is travelling in the opposite direction. However, it takes 20 seconds, if both the trains are travelling in the same direction. The length of the first train is 200 m and that of the second train is 150 m. What is the speed of the first train ?
54 km/hr
Solving Train Speed Problems with Relative Speed
This problem involves two trains moving in opposite and same directions, and we need to find the speed of the first train based on the time it takes to cross a man in the second train. This requires understanding the concept of relative speed and how distance is defined in such scenarios.
Let's define the given information and variables:
- Length of the first train, \(L_1 = 200\) m
- Length of the second train, \(L_2 = 150\) m
- Time taken to cross a man in the other train when travelling in opposite directions, \(t_{opp} = 10\) seconds
- Time taken to cross a man in the other train when travelling in the same direction, \(t_{same} = 20\) seconds
- Let \(S_1\) be the speed of the first train (in m/s).
- Let \(S_2\) be the speed of the second train (in m/s).
Understanding Relative Speed and Distance
When a train crosses a stationary object (like a pole, a signal, or a man standing still), the distance covered is the length of the train itself, and the speed is the train's own speed. When a train crosses a man travelling in another train, the situation is slightly different because the man is also moving. The distance covered by the first train in crossing the man is still the length of the first train (\(L_1\)), but the speed to consider is the relative speed of the first train with respect to the man.
Since the man is travelling in the second train, his speed relative to the ground is the speed of the second train, \(S_2\).
Case 1: Trains Travelling in Opposite Directions
When the trains move in opposite directions, their speeds add up to give the relative speed between them. The relative speed of the first train with respect to the man (who is moving at \(S_2\)) is the sum of their speeds:
Relative speed = \(S_1 + S_2\)
The distance covered by the first train in crossing the man is the length of the first train, \(L_1\).
Using the formula Distance = Speed \(\times\) Time, we have:
\[L_1 = (S_1 + S_2) \times t_{opp}\] \[200 = (S_1 + S_2) \times 10\]Dividing both sides by 10:
\[S_1 + S_2 = \frac{200}{10}\] \[S_1 + S_2 = 20 \text{ m/s} \quad \text{(Equation 1)}\]Case 2: Trains Travelling in the Same Direction
When the trains move in the same direction, the relative speed between them is the difference between their speeds. Assuming the first train is faster than the second train (which must be true for it to cross the man in the second train when moving in the same direction), the relative speed of the first train with respect to the man is:
Relative speed = \(S_1 - S_2\)
The distance covered by the first train in crossing the man is still the length of the first train, \(L_1\).
Using the formula Distance = Speed \(\times\) Time, we have:
\[L_1 = (S_1 - S_2) \times t_{same}\] \[200 = (S_1 - S_2) \times 20\]Dividing both sides by 20:
\[S_1 - S_2 = \frac{200}{20}\] \[S_1 - S_2 = 10 \text{ m/s} \quad \text{(Equation 2)}\]Solving for the Speed of the First Train
Now we have a system of two linear equations with two variables \(S_1\) and \(S_2\):
- \(S_1 + S_2 = 20\)
- \(S_1 - S_2 = 10\)
We can solve this system by adding the two equations:
\[(S_1 + S_2) + (S_1 - S_2) = 20 + 10\] \[S_1 + S_2 + S_1 - S_2 = 30\] \[2S_1 = 30\]Now, divide by 2 to find \(S_1\):
\[S_1 = \frac{30}{2}\] \[S_1 = 15 \text{ m/s}\]So, the speed of the first train is 15 m/s. We can also find the speed of the second train by substituting \(S_1 = 15\) into Equation 1:
\[15 + S_2 = 20\] \[S_2 = 20 - 15\] \[S_2 = 5 \text{ m/s}\]This confirms our assumption that \(S_1 > S_2\) (15 m/s > 5 m/s).
Converting Speed to km/hr
The options are given in km/hr, so we need to convert the speed of the first train from m/s to km/hr. The conversion factor is \(\frac{18}{5}\) (since 1 km = 1000 m and 1 hour = 3600 seconds, so 1 m/s = \(\frac{1/1000}{1/3600}\) km/hr = \(\frac{3600}{1000}\) km/hr = \(\frac{18}{5}\) km/hr).
\[S_1 \text{ in km/hr} = 15 \text{ m/s} \times \frac{18}{5} \text{ km/hr per m/s}\] \[S_1 = 3 \times 18\] \[S_1 = 54 \text{ km/hr}\]The speed of the first train is 54 km/hr.
Revision Table: Key Formulas for Train Problems
| Concept | Formula | Notes |
|---|---|---|
| Speed, Distance, Time | \(D = S \times T\) | \(S = D/T\), \(T = D/S\) |
| Relative Speed (Opposite Direction) | \(S_{rel} = S_A + S_B\) | Speeds add up |
| Relative Speed (Same Direction) | \(S_{rel} = |S_A - S_B|\) | Difference in speeds |
| Train crossing a point object (pole, man) | \(D = \text{Length of Train}\) | Speed is relative speed of train w.r.t object |
| Train crossing another train | \(D = \text{Length}_1 + \text{Length}_2\) | Speed is relative speed of one train w.r.t other train |
| Convert m/s to km/hr | Multiply by \(\frac{18}{5}\) | \(1 \text{ m/s} = \frac{18}{5} \text{ km/hr}\) |
Additional Information: Interpreting "Crossing a Man"
It's common in train problems to encounter scenarios like a train crossing a pole, a platform, or another train. The phrasing "crossing a man travelling in another train" is key here. If the question had said "the time taken by the first train to cross the second train", then the distance would indeed be the sum of their lengths (\(L_1 + L_2\)). However, by specifying "crossing a man", the distance being covered by the first train relative to the moving man is its own length. Think of it from the perspective of the man; he sees the entire length of the first train pass by him.
The relative speed calculation correctly accounts for the man's movement because his speed is tied to the speed of the second train. Whether the trains are moving towards each other or away from each other (or one overtaking the other) dictates how their speeds combine for the relative speed calculation, while the distance is fixed as the length of the train doing the crossing (the first train, in this case).
This problem effectively tests the ability to correctly identify the relevant distance and relative speed based on the specifics of what is being crossed (a man vs. an entire train).
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Important Questions from Problem on Trains
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A daily train is to be introduced between station A and station B starting from each end at 6 AM and the journey is to be completed in 42 hours. What is the number of trains needed in order to maintain the Shuttle Service?
A train with a uniform speed passes a 122 meters long platform in 17 seconds and a 210 meters long bridge in 25 seconds. The speed of the train is:
How long does a train 153 meters long running at the rate of 90 kmph take to cross a bridge 622 meters in length?
A train passes a 360 metre long platform in 40 seconds and a man standing on the platform in 16 seconds. The speed of the train is:
