 # Train Problems

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# Train Problems

### Introduction

Train problems are totally based on four topics including conversion, distance formula, relativity, and train theory.

### Methods

Conversion: It includes conversion of kilometer per hour (kmph) into meter per second (mps) or vice -versa.
Distance formula: D = S x T where,
D ⇒ Distance S ⇒ Speed T ⇒ Time

Relativity: It is a broad term. It describes about objects moving in either direction or in the same direction, speed of the objects, and time taken by the objects, etc.
Example 1: A train 100 m long is running at the speed of 30 km/hr. find the time taken by in to pass a man standing near the railway line.
Solution:
Speed of the train = (30 x $\frac{5}{8}$) m/sec = ($\frac{25}{3}$) m/sec
Distance moved in passing the standing man = 100 m.
Required time taken = ($\frac{100}{\frac{25}{3}}$) = (100 x $\frac{3}{25}$) sec = 12 sec.
Example 2: A train is moving at a speed of 132 km/hr. If the length of the train is 110 metres, how long will it take to cross a railway platform 165 metre long?
Solution:
Speed of the train = (132 x $\frac{5}{18}$) m/sec = ($\frac{110}{3}$) m/sec
Distance covered in passing the platform = (110+165) m = 275 m.
Time taken = (275 x $\frac{3}{110}$) sec = ($\frac{15}{2}$) sec = 7$\frac{1}{2}$ sec

Example 1: A man is standing on a railway bridge which is 180 m long. He finds that a train crosses the bridge in 20 seconds but himself in 8 seconds. Find the length of the train and its speed.
Solution:
Let the length of the train be x metres.
Then, the train covers x metres in 8 seconds and ($x$ + 180) metres in 20 seconds.
∴ $\frac{x}{8}$ = $\frac{(x + 180)}{20}$ ⇔ 20$x$ = 8($x$ + 180) ⇔ $x$ = 120.
∴ Length of the train = 120 m.
Speed of the train = ($\frac{120}{8}$) m/sec = m/sec = (15 x $\frac{18}{5}$) kmph = 54 kmph.

Example 2: A man sitting in a train which is travelling at 50 kmph observes that a goods train, travelling in opposite direction, takes 9 seconds to pass him. if the goods train is 280 m long, find its speed.
Solution:
Relative speed = ($\frac{280}{9}$) m/sec = ($\frac{280}{9}$ x $\frac{18}{5}$) kmph = 112 kmph.
∴ Speed of goods train = (112 - 50) kmph = 62 kmph.

### Formulae

1. $x$ km/hr = $x$ x $\frac{5}{18}$ m/s
2. $x$ m/s = $x$ x $\frac{18}{5}$ km/hr
3. Time taken by a train of length $y$ meters to pass a pole or a standing man or a signal post or an object of negligible width would be equal to the time taken by the train to cover $y$ meters which is primarily the length of the train.
4. Time taken by a train of length $y$ meters to pass a stationary object of length $b$ meters is the time taken by the train to cover ($y + b$) meters.
5. Suppose two trains or two bodies are moving in the same direction at $x$ m/s and $y$ m/s, where $x$ > $y$, then their Relative speed = ($x$ - $y$) m/s
6. Suppose two trains or two bodies are moving in the opposite direction at $x$ m/s and $y$ m/s, then their Relative speed = ($x$ + $y$) m/s
7. If two trains of length $a$ meters and $b$ meters are moving in the opposite direction at $x$ m/s $y$ m/s, then the time taken by the faster train to cross the slower train = $\frac{(a + b)}{(x + y)}$ sec.
8. If two trains of length $a$ meters and $b$ meters are moving in the same direction at $x$ m/s $y$ m/s, then the time taken by the faster train to cross the slower train = $\frac{(a + b)}{(x - y)}$ sec.
9. If the two trains (or bodies) start at the same time from points A and B towards each other and after crossing they take $x$ and $y$ sec in reaching B and A respectively, then (A's speed) : (B's speed) = $\sqrt{y}$ : $\sqrt{x}$

### Samples

1. A train of length 200 m is running at the speed of 20 km/hr. Find the time taken by the train to pass a man standing near the railway line?
Solution:
Given,
Distance moved in passing the standing man = 200 m
Speed = 20 kmph
Convert kmph to mps
Speed = 20 x $\frac{5}{18}$ = $\frac{50}{9}$
So, Required time = $\frac{distance}{speed}$
⇒ Time = $\frac{200}{\frac{50}{9}}$
⇒ Time = $\frac{200 * 9}{50}$
⇒ Time = 36 sec
Therefore, the time taken by it to pass a man standing near the railway line = 36 sec

2. A man is standing on a railway bridge which is 200m long. He finds that a train crosses the bridge in 10 seconds but himself in 6 seconds. Find the length of the train and its speed?
Solution:
Let,
The length of the train = $x$ m
Then, the train covers $x$ m in 6 seconds and $x + 200$ m in 10 seconds.
So, Length of the train is
$\frac{x}{6}$ = $\frac{x + 200}{10}$
⇒ 10 x $x$ = 6($x$ + 200)
⇒ 10 x $x$ - 6 x $x$ = 1200
⇒ 4 x $x$ = 1200
⇒ $x$ = 300
⇒ Length of train = 300 m
Then, Speed of the train = $\frac{300}{6}$ m/s = 50 m/s
⇒ 50 x $\frac{18}{5}$ km/hr
⇒ 180 km/hr
Therefore, length of train = 300m and
Speed of train = 180 km/hr.

3. A train 320 m long is running with speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the opposite direction in which the train is going?
Solution:
Given,
Relative Speed of the train and man = (60 + 6) = 66 kmph
Converting it into m/s, i.e.
⇒ 66 x $\frac{5}{18}$
⇒ $\frac{55}{3}$ m/s
Time taken by the train to cross the man is
= time taken by it to cover 320 m at $\frac{55}{3}$ m/s
= 320 x $\frac{3}{55}$
= 17.45 sec
Therefore, time taken by the train to cross the man is 17.45 sec

4. Two trains 200 m and 320 m long are running in the same direction with speeds of 72 km/hr and 54 km/hr. In how much time will the first train cross the second?
Solution:
Relative Speed of the trains = (72 - 54) km/hr = 18 km/hr
Convert it into m/sec, i.e.
⇒ 18 x $\frac{5}{18}$
⇒ 5 m/s
Time taken by the trains to cross each other is
= time taken to cover (200 + 320)m at 5 m/s
= $\frac{520}{5}$
= 104
Therefore, time taken by the trains to cross each other is 104 sec.

5. A man sitting in a train which is travelling at 60 kmph observes that a goods train, travelling in opposite direction, takes 9 seconds to pass him. If the goods train is 300 m long, find its speed?
Solution:
Given,
The goods train travels 60 kmph
Relative speed = $\frac{300}{9}$ m/s
Convert it into kmph, i.e.
$\frac{300}{9}$ x $\frac{18}{5}$ = 120kmph
Now, Speed of goods train is
= (120 - 60) kmph
= 60 kmph
Therefore, Speed of goods train = 60 kmph