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:

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

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:

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

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

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

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:

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

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

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