Problems on Trains

The fundamental formula relating speed, distance, and time is:

$$ \text{Distance} = \text{Speed} \times \text{Time} $$

Shorthand: \(D = S \times T\)

Given:

• Speed, \( S = 60 \) km/h
• Time, \( T = 3 \) hours

Using the formula:

$$ \text{Distance} = \text{Speed} \times \text{Time} $$

Substitute the given values:

$$ \text{Distance} = 60 \, \text{km/h} \times 3 \, \text{hours} $$

Calculate:

$$ \text{Distance} = 180 \, \text{km} $$

To convert speed from km/hr to m/s and vice versa:

$$ 1 \text{ km/hr} = \frac{5}{18} \text{ m/s} $$

$$ 1 \text{ m/s} = \frac{18}{5} \text{ km/hr} $$

When a train crosses a stationary object (like a pole) or a person, the distance covered is equal to the length of the train:

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

When two trains are moving in the same direction:

$$ \text{Relative Speed} = \left| \text{Speed of Train 1} - \text{Speed of Train 2} \right| $$

When two trains are moving in opposite directions:

$$ \text{Relative Speed} = \text{Speed of Train 1} + \text{Speed of Train 2} $$

When two trains of lengths \( L_1 \) and \( L_2 \) cross each other, the time taken to cross is:

$$ \text{Time} = \frac{L_1 + L_2}{\text{Relative Speed}} $$

When a train of length \( L \) crosses a platform of length \( P \), the distance covered is the sum of the lengths of the train and the platform:

$$ \text{Distance} = L + P $$

So, the time taken to cross the platform is:

$$ \text{Time} = \frac{L + P}{\text{Speed}} $$

Problem: A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train?

Solution:

$$ \text{Speed in m/s} = 60 \times \frac{5}{18} = 16.67 \text{ m/s} $$

$$ \text{Length of the Train} = \text{Speed} \times \text{Time} = 16.67 \times 9 = 150 \text{ meters} $$

Problem: Two trains are moving in opposite directions at speeds of 54 km/hr and 36 km/hr. If the length of each train is 150 meters, how long will it take for them to cross each other?

Solution:

$$ \text{Relative Speed in m/s} = \left(54 + 36\right) \times \frac{5}{18} = 25 \text{ m/s} $$

$$ \text{Total Distance} = 150 + 150 = 300 \text{ meters} $$

$$ \text{Time} = \frac{\text{Total Distance}}{\text{Relative Speed}} = \frac{300}{25} = 12 \text{ seconds} $$

Problem: A train of length 200 meters crosses a platform of length 300 meters in 30 seconds. What is the speed of the train in km/hr?

Solution:

$$ \text{Total Distance} = 200 + 300 = 500 \text{ meters} $$

$$ \text{Speed in m/s} = \frac{\text{Total Distance}}{\text{Time}} = \frac{500}{30} = 16.67 \text{ m/s} $$

$$ \text{Speed in km/hr} = 16.67 \times \frac{18}{5} = 60 \text{ km/hr} $$