A train is travelling at 48 km/hr completely crosses another train having half its length and travelling in opposite direction at 42 km/hr in 12 s. It also passes a railway platform in 45 s. What is the length of the platform?
400 m
Solving Train Speed and Length Problems
This problem involves calculating the length of a railway platform using information about two trains and their crossing times. We need to use the fundamental relationship between speed, distance, and time, considering different scenarios for crossing.
The key principle for solving train problems is understanding the distance covered when a train crosses an object. When a train crosses:
- A point (like a pole or a person), the distance covered is the train's own length.
- An object with length (like another train or a platform), the distance covered is the sum of the train's length and the object's length.
Also, when two trains move relative to each other, their relative speed is used:
- If they move in the same direction, relative speed is the difference between their speeds.
- If they move in opposite directions, relative speed is the sum of their speeds.
Step-by-Step Solution for the Train Problem
Let's break down the problem into the two given scenarios and use the information to find the unknown lengths.
Understanding the Given Information
We have the following information about the trains:
- Speed of the first train (\(\text{V}_1\)): 48 km/hr
- Length of the second train is half the length of the first train.
- Speed of the second train (\(\text{V}_2\)): 42 km/hr
- Direction of the second train: Opposite to the first train.
- Time taken for the two trains to cross each other: 12 seconds.
- Time taken for the first train to cross a platform: 45 seconds.
Converting Units
Speeds are given in km/hr and time in seconds. To use the formula Distance = Speed \(\times\) Time consistently in meters and seconds, we must convert the speeds from km/hr to m/s. The conversion factor is \(\frac{5}{18}\).
- Speed of first train (\(\text{V}_1\)): \(48 \text{ km/hr} = 48 \times \frac{5}{18} \text{ m/s} = \frac{8 \times 5}{3} \text{ m/s} = \frac{40}{3} \text{ m/s}\)
- Speed of second train (\(\text{V}_2\)): \(42 \text{ km/hr} = 42 \times \frac{5}{18} \text{ m/s} = \frac{7 \times 5}{3} \text{ m/s} = \frac{35}{3} \text{ m/s}\)
Scenario 1: Two Trains Crossing
Let the length of the first train be \(L_1\) meters. The length of the second train is \(L_2 = \frac{L_1}{2}\) meters.
When the two trains cross each other while travelling in opposite directions, the total distance covered is the sum of their lengths (\(L_1 + L_2\)), and the relative speed is the sum of their speeds (\(\text{V}_1 + \text{V}_2\)).
- Total distance: \(L_1 + L_2 = L_1 + \frac{L_1}{2} = \frac{3L_1}{2}\) meters.
- Relative speed: \(\text{V}_1 + \text{V}_2 = \frac{40}{3} + \frac{35}{3} = \frac{75}{3} = 25 \text{ m/s}\).
- Time taken: 12 seconds.
Using the formula Distance = Speed \(\times\) Time:
\(\frac{3L_1}{2} = 25 \text{ m/s} \times 12 \text{ s}\)
\(\frac{3L_1}{2} = 300\)
\(3L_1 = 300 \times 2\)
\(3L_1 = 600\)
\(L_1 = \frac{600}{3}\)
\(L_1 = 200 \text{ meters}\)
So, the length of the first train is 200 meters.
Scenario 2: First Train Crossing a Platform
The first train (length \(L_1 = 200\) m, speed \(\text{V}_1 = \frac{40}{3}\) m/s) crosses a railway platform in 45 seconds. Let the length of the platform be \(P\) meters.
When the first train crosses the platform, the total distance covered is the sum of the train's length and the platform's length (\(L_1 + P\)). The speed used is the speed of the train.
- Total distance: \(L_1 + P = 200 + P\) meters.
- Speed of the first train: \(\text{V}_1 = \frac{40}{3} \text{ m/s}\).
- Time taken: 45 seconds.
Using the formula Distance = Speed \(\times\) Time:
\(200 + P = \frac{40}{3} \text{ m/s} \times 45 \text{ s}\)
\(200 + P = 40 \times \frac{45}{3}\)
\(200 + P = 40 \times 15\)
\(200 + P = 600\)
\(P = 600 - 200\)
\(P = 400 \text{ meters}\)
The length of the platform is 400 meters.
Let's summarize the findings:
| Item | Length (m) | Speed (m/s) |
|---|---|---|
| First Train | 200 | \( \frac{40}{3} \) |
| Second Train | 100 (half of 200) | \( \frac{35}{3} \) |
| Platform | 400 | N/A (stationary) |
The length of the platform is 400 meters.
Revision Table: Key Concepts for Train Problems
| Scenario | Distance Covered | Relative Speed (Opposite Direction) | Relative Speed (Same Direction) | Formula |
|---|---|---|---|---|
| Train crosses Point/Pole/Person | Length of Train | Speed of Train | Speed of Train | Distance = Speed \(\times\) Time |
| Train crosses Stationary Object (Platform, Bridge) | Length of Train + Length of Object | Speed of Train | Speed of Train | Distance = Speed \(\times\) Time |
| Train 1 crosses Train 2 (Opposite Direction) | Length of Train 1 + Length of Train 2 | Speed of Train 1 + Speed of Train 2 | N/A | Distance = Relative Speed \(\times\) Time |
| Train 1 crosses Train 2 (Same Direction) | Length of Train 1 + Length of Train 2 | N/A | |Speed of Train 1 - Speed of Train 2| | Distance = Relative Speed \(\times\) Time |
Additional Information on Train and Platform Calculations
Train problems are a common topic in quantitative aptitude. They test your understanding of the relationship between distance, speed, and time, especially when dealing with moving objects or objects with considerable length. Always ensure all units are consistent (e.g., meters and seconds) before performing calculations. Converting speeds from km/hr to m/s using the \(\frac{5}{18}\) factor is crucial. Remember that when a train crosses an object of length, the total distance is the sum of the lengths. Relative speed is used only when both objects are moving.
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Important Questions from Problem on Trains
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If the route between G and C is closed, which one of the following stations need not be passed through while travelling from H to C?
A daily train is to be introduced between station A and station B starting from each end at 6 AM and the journey is to be completed in 42 hours. What is the number of trains needed in order to maintain the Shuttle Service?
A train with a uniform speed passes a 122 meters long platform in 17 seconds and a 210 meters long bridge in 25 seconds. The speed of the train is:
How long does a train 153 meters long running at the rate of 90 kmph take to cross a bridge 622 meters in length?
A train passes a 360 metre long platform in 40 seconds and a man standing on the platform in 16 seconds. The speed of the train is:
