Train Problems – Time and Distance // Type 5: Type 6: Type 7: Type 8: Type 9

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TYPE 5: Crossing a Moving Man (Same Direction)

Definition: When a train and a man are moving in the same direction, use relative speed.

$$\text{Relative Speed }=\text{Train Speed }-\text{Man’s Speed}$$

Q. A 150-metre-long train is moving at 50 km/h. A man is running in the same direction at 5 km/h. In how much time will the train cross the man ?

Formula: Time = Length of Train / Relative Speed​

Solution:

Train's speed = 50 km/h
Man's speed = 5 km/h

Relative speed: $$50-5=45\text{ km/h}$$

Convert into m/s: 45 × 5​/18 = 12.5 m/s

Now,

Time =150​/12.5 = 12 seconds​

Ans: 12 seconds

Trick: i. Same direction → Subtract speeds   ii. Opposite direction → Add speeds

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TYPE 6: Crossing a Moving Man (Opposite Direction)

Definition: When a train and a man are moving in opposite directions, their relative speed is the sum of their speeds.

$$\text{Relative Speed }=\text{Train Speed }+\text{Man’s Speed}$$

Q. A 100-metre-long train is moving at 30 km/h. A man is coming towards the train at 6 km/h. In how much time will the train cross the man ?

Formula: Time = Length of Train / Relative Speed

​Solution:

Train's speed = 30 km/h
Man's speed = 6 km/h

Relative speed: $$30+6=36\text{ km/h}$$

Convert into m/s: 36 × 5​/18 = 10 m/s

Now,

Time = 100/10​ = 10 seconds​

Ans: 10 seconds

Trick: i. Same direction → Subtract speeds,  ii. Opposite direction → Add speeds

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TYPE 7: Two Trains Crossing Each Other (Opposite Direction)

Definition: When two trains are moving in opposite directions, the time taken to cross each other is:

Time = Total Distance / Relative Speed​

Where:

Relative Speed = Speed of Train 1 + Speed of Train 2

Q. Two trains of lengths 120 m and 80 m are moving in opposite directions at 40 km/h and 50 km/h, respectively. How much time will they take to cross each other ?

Formula: Time = Length of Train 1 + Length of Train 2 / Relative Speed​

Solution:

Train 1 length = 120 m
Train 2 length = 80 m

Total distance: $$120+80=200\text{ m}$$

Relative speed: $$40+50=90\text{ km/h}$$

Convert into m/s: 90 × 185​ = 25 m/s

Now,

Time = 200​/25 = 8 seconds​

Ans: 8 seconds

Trick: i. Opposite direction → Add speeds,  Total distance → Add train lengths

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TYPE 8: Two Trains Crossing Each Other (Same Direction)

Definition: When two trains are moving in the same direction, the faster train completely crosses the slower train using their relative speed.

Q. Two trains of lengths 100 m and 80 m are moving in the same direction at 54 km/h and 36 km/h, respectively. How much time will the faster train take to completely cross the slower train ?

Formula: Time  =  Total Distance / Relative Speed​

For the same direction:

$$\text{Relative Speed }=\text{Faster Speed }-\text{Slower Speed}$$

Solution:

Total distance: $$100+80=180\text{ m}$$

Relative speed: $$54-36=18\text{ km/h}$$

Convert into m/s: 18 × 5/18 ​= 5 m/s

Now,

Time = 180​/5 = 36 seconds​

Ans: 36 seconds

Trick: i. Same direction → Subtract speeds,  ii. Opposite direction → Add speeds,  iii. Two trains → Add their lengths

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TYPE 9: Crossing a Person Sitting in Another Train

Definition: When a train crosses a person sitting inside another moving train, the distance covered is only the length of the moving train that is crossing the person.

Formula: Time = Distance / Relative Speed​

Therefore,

$$\text{Length of Train }=\text{Relative Speed }\times \text{Time}$$

Q. Two trains are moving in opposite directions at 72 km/h and 36 km/h. The faster train crosses a person sitting in the slower train in 5 seconds. What is the length of the faster train ?

Solution:

Train 1 speed = 72 km/h
Train 2 speed = 36 km/h

Since they are moving in opposite directions:

$$\text{Relative Speed }=72+36=108\text{ km/h}$$

Convert into m/s: 108 × 185 ​= 30 m/s

Time = 5 seconds

Therefore,

$$\text{Length of faster train }=30\times 5$$ = 150 m​

Ans: 150 metres

Trick: When crossing a person sitting in another train → Distance = Length of the crossing train only.

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