Arithmetic

Time, Speed & Distance

Medium

Two persons are walking beside a railway track at respective speeds of $2$ and $4 \mathrm{~km}$ per hour in the same direction. A train came from behind them and crossed them in $90$ and $100$ seconds, respectively. The time, in seconds, taken by the train to cross a stationary object?

$78$

$87$

$75$

$82$

✅ Correct Option: 4

We have two people walking alongside a railway track, and a train coming from behind crosses both of them. We need to find how long it takes for the train to cross a stationary object.

Given information:

Person 1 walks at 2 km/h

Person 2 walks at 4 km/h

Train crosses Person 1 in 90 seconds

Train crosses Person 2 in 100 seconds

When a train crosses a moving person, we use relative speed because both objects are moving.

Relative Speed Formula:

When moving in the same direction: Relative Speed = Train Speed - Person Speed

When moving in opposite directions: Relative Speed = Train Speed + Person Speed

Since the train comes from behind (same direction), we subtract the person's speed from the train's speed.

Let the train's speed be S km/h.

For Person 1 (2 km/h):

Relative speed = (S - 2) km/h

Time taken = 90 seconds

Distance covered = Length of train

For Person 2 (4 km/h):

Relative speed = (S - 4) km/h

Time taken = 100 seconds

Distance covered = Length of train (same as above)

Since the train length remains constant, we can set up an equation:

Distance = Speed × Time

For both crossings, the distance (train length) is the same:

$(S - 2) \times 90 = (S - 4) \times 100$

Note: We don't need to convert units here since we're finding a ratio.

$90S - 180 = 100S - 400$

$400 - 180 = 100S - 90S$

$220 = 10S$

$S = 22 \text{ km/h}$

Using the train speed in the first scenario:

Relative speed = 22 - 2 = 20 km/h

Time = 90 seconds = $\tfrac{90}{3600}$ hours = $\tfrac{1}{40}$ hours

Train length = Speed × Time

$\text{Train length} = 20 \times \tfrac{1}{40} = 0.5 \text{ km}$

When crossing a stationary object, there's no relative motion to consider.

Time = Distance ÷ Speed

$\text{Time} = \tfrac{0.5 \text{ km}}{22 \text{ km/h}} = \tfrac{0.5}{22} \text{ hours}$

Converting to seconds:

$\tfrac{0.5}{22} \times 3600 = \tfrac{1800}{22} = 81.81 \text{ seconds}$

Therefore, the train takes approximately 82 seconds to cross a stationary object.

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Two persons are walking beside a railway track at respective speeds of 222 and 4 km4 \mathrm{~km}4 km per hour in the same direction. A train came from behind them and crossed them in 909090 and 100100100 seconds, respectively. The time, in seconds, taken by the train to cross a stationary object?

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Two persons are walking beside a railway track at respective speeds of $2$ and $4 \mathrm{~km}$ per hour in the same direction. A train came from behind them and crossed them in $90$ and $100$ seconds, respectively. The time, in seconds, taken by the train to cross a stationary object?