Trains

1. BASIC CONCEPT

Train problems are an important application of Time, Distance and Speed. The main idea is to determine the total distance that a train must travel and divide it by the appropriate speed.

``` Basic Formula:

Speed = Distance ÷ Time
Distance = Speed × Time
Time = Distance ÷ Speed ```

``` Important quantities in train problems:

Length of train: Usually given in metres.
Speed: Usually given in km/h or m/s.
Time: Usually required in seconds.
Platform: Its length must be added to train length when the train completely crosses it.

```

2. SPEED CONVERSION

``` 1 km = 1000 m
1 hour = 3600 seconds

km/h → m/s = × 5/18
m/s → km/h = × 18/5 ```

``` Example 1: Convert 72 km/h into m/s.

72 × 5/18 = 20 m/s. ```

``` Example 2: Convert 25 m/s into km/h.

25 × 18/5 = 90 km/h. ```

3. TRAIN CROSSING A POLE

When a train crosses a pole, person or stationary object, the train has to cover a distance equal to its own length.

``` Time = Length of train ÷ Speed

Length of train = Speed × Time

Speed = Length of train ÷ Time ```

``` Example 3: A train 180 m long travels at 54 km/h. Find the time taken to cross a pole.

54 km/h = 54 × 5/18 = 15 m/s
Time = 180/15 = 12 seconds. ```

``` Example 4: A train crosses a pole in 10 seconds at 72 km/h. Find its length.

72 km/h = 20 m/s
Length = 20 × 10 = 200 m. ```

``` Shortcut: Pole, tree, person or signal post means only the train's own length is considered. ```

4. TRAIN CROSSING A PLATFORM

To completely cross a platform, the train must travel a distance equal to the sum of the train's length and the platform's length.

``` Distance travelled = Train length + Platform length

Time = (Train length + Platform length) ÷ Speed ```

``` Example 1: A train 150 m long crosses a platform 250 m long at 72 km/h. Find the time taken.

Total distance = 150 + 250 = 400 m
72 km/h = 20 m/s
Time = 400/20 = 20 seconds. ```

``` Example 2: A train 200 m long crosses a platform in 20 seconds at 54 km/h. Find the platform length.

54 km/h = 15 m/s
Total distance = 15 × 20 = 300 m
Platform length = 300 − 200 = 100 m. ```

5. TRAIN CROSSING A BRIDGE OR TUNNEL

The same principle applies to a bridge, tunnel or any other fixed structure.

``` Time = (Length of train + Length of structure) ÷ Speed ```

``` Example 3: A 250 m train crosses a 350 m bridge at 60 km/h.

Total distance = 250 + 350 = 600 m
60 km/h = 50/3 m/s
Time = 600 ÷ (50/3) = 600 × 3/50 = 36 seconds. ```

6. RELATIVE SPEED

When two trains are moving, their relative speed determines how quickly they cross each other.

``` Opposite directions: Relative speed = S₁ + S₂

Same direction: Relative speed = |S₁ − S₂| ```

``` Example 4: Two trains move towards each other at 60 km/h and 40 km/h.

Relative speed = 60 + 40 = 100 km/h. ```

``` Example 5: Two trains move in the same direction at 80 km/h and 50 km/h.

Relative speed = 80 − 50 = 30 km/h. ```

7. TWO TRAINS CROSSING EACH OTHER

``` Time = Sum of train lengths ÷ Relative speed ```

``` Example 6: Two trains of lengths 200 m and 300 m move in opposite directions at 54 km/h and 36 km/h.

Total length = 200 + 300 = 500 m
Relative speed = 54 + 36 = 90 km/h
90 km/h = 25 m/s
Time = 500/25 = 20 seconds. ```

``` Exam Rule: When two trains completely cross each other, add their lengths because both trains must pass completely. ```

8. TWO TRAINS MOVING IN THE SAME DIRECTION

When a faster train overtakes a slower train moving in the same direction, use the difference of their speeds.

``` Relative speed = Faster speed − Slower speed

Time = Sum of lengths ÷ Relative speed ```

``` Example 1: A train 180 m long and another train 120 m long move in the same direction at 72 km/h and 54 km/h. Find the time taken by the faster train to completely overtake the slower train.

Total length = 180 + 120 = 300 m
Relative speed = 72 − 54 = 18 km/h
18 km/h = 5 m/s
Time = 300/5 = 60 seconds. ```

9. TRAIN CROSSING A MAN MOVING

If a person is walking or running, the relative speed must be used instead of the train's actual speed.

``` Same direction: Relative speed = Train speed − Man speed

Opposite direction: Relative speed = Train speed + Man speed ```

``` Example 2: A train runs at 72 km/h and a man runs in the same direction at 18 km/h. If the train is 270 m long, find the time taken to cross the man.

Relative speed = 72 − 18 = 54 km/h
54 km/h = 15 m/s
Time = 270/15 = 18 seconds. ```

10. TRAIN CROSSING A MAN ON A PLATFORM

If the person is stationary, simply use the train's speed and train's length. If the person is moving, use relative speed.

``` Example 3: A train 240 m long runs at 54 km/h. A person runs towards the train at 6 km/h. Find the time required for the train to cross him.

Relative speed = 54 + 6 = 60 km/h
60 km/h = 50/3 m/s
Time = 240 ÷ (50/3) = 240 × 3/50 = 14.4 seconds. ```

11. TRAIN SPEED FROM LENGTH AND CROSSING TIME

``` Speed = Distance ÷ Time

For pole: Speed = Train length ÷ Time

For platform: Speed = (Train length + Platform length) ÷ Time ```

``` Example 4: A train 300 m long crosses a pole in 15 seconds. Find its speed in km/h.

Speed = 300/15 = 20 m/s
Speed = 20 × 18/5 = 72 km/h. ```

12. RATIO METHOD

``` For the same train and same distance:

Speed ∝ 1/Time
Therefore, S₁/S₂ = T₂/T₁ ```

``` Example 5: A train takes 20 seconds to cross a pole at 54 km/h. How much time will it take at 72 km/h?

T₂ = T₁ × S₁/S₂
T₂ = 20 × 54/72
T₂ = 15 seconds. ```

13. IMPORTANT CASES AT A GLANCE

``` ```

Situation	Distance to be Covered
Train + Pole	Train length
Train + Person standing	Train length
Train + Platform	Train length + Platform length
Train + Bridge	Train length + Bridge length
Two trains	Sum of their lengths

14. COMPLETE FORMULA SHEET

``` ```

Concept	Formula
Speed	Distance ÷ Time
Time	Distance ÷ Speed
Train crossing pole	L/S
Train crossing platform	(L + P)/S
Two trains opposite	(L₁ + L₂)/(S₁ + S₂)
Two trains same direction	(L₁ + L₂)/\|S₁ − S₂\|
Train & moving person	Train length ÷ Relative speed
km/h → m/s	× 5/18
m/s → km/h	× 18/5

15. SOLVED REVISION QUESTIONS

``` Example 1: A train 240 m long travels at 72 km/h. How long will it take to cross a pole?

72 km/h = 20 m/s
Time = 240/20 = 12 seconds. ```

``` Example 2: A train 180 m long crosses a 220 m platform at 72 km/h.

Total distance = 180 + 220 = 400 m
Speed = 20 m/s
Time = 400/20 = 20 seconds. ```

``` Example 3: Two trains of lengths 150 m and 250 m move in opposite directions at 54 km/h and 36 km/h.

Relative speed = 90 km/h = 25 m/s
Total length = 400 m
Time = 400/25 = 16 seconds. ```

``` Example 4: Two trains of lengths 200 m and 100 m move in the same direction at 72 km/h and 54 km/h.

Relative speed = 18 km/h = 5 m/s
Total length = 300 m
Time = 300/5 = 60 seconds. ```

16. PRACTICE QUESTIONS

A train 180 m long runs at 54 km/h. Find the time taken to cross a pole.
A train 250 m long runs at 72 km/h. Find the time taken to cross a pole.
A train 200 m long crosses a 300 m platform at 90 km/h. Find the time taken.
A train crosses a pole in 15 seconds at 72 km/h. Find the length of the train.
Two trains of lengths 150 m and 250 m move in opposite directions at 54 km/h and 36 km/h. Find their crossing time.
Two trains of lengths 200 m and 100 m move in the same direction at 72 km/h and 54 km/h. Find their crossing time.
A train 300 m long crosses a platform 200 m long in 25 seconds. Find its speed.
A train runs at 90 km/h and crosses a person in 8 seconds. Find the train's length.
A 240 m train runs at 54 km/h. A man runs in the opposite direction at 6 km/h. Find the time taken by the train to cross the man.
A train takes 20 seconds to cross a pole at 54 km/h. Find the time it will take at 72 km/h.

17. ANSWERS

``` ```

Q	Answer	Q	Answer
1	12 seconds	6	60 seconds
2	12.5 seconds	7	72 km/h
3	20 seconds	8	200 m
4	300 m	9	18 seconds
5	16 seconds	10	15 seconds

18. EXAM SHORTCUTS

``` Shortcut 1: Pole/person standing → use only train length. ```

``` Shortcut 2: Platform/bridge/tunnel → add train length and structure length. ```

``` Shortcut 3: Opposite direction → add speeds. ```

``` Shortcut 4: Same direction → subtract speeds. ```

``` Shortcut 5: If lengths are in metres and time is in seconds, convert speed into m/s before calculation. ```

``` Common Mistake: Do not use km/h directly with metre and second values. Convert km/h to m/s using × 5/18. ```

``` UTKALPRESS — ARITHMETIC SERIES

Remember:
Distance = Speed × Time
Train problems = Distance + Relative Speed
Understand the situation first, then select the formula. ```