A train of length 110 m is moving at a uniform of 132 km/hr. The time required to cross a bridge of length 165 m is
7.5 seconds
Understanding the Problem: Train Crossing a Bridge
This problem asks us to calculate the time it takes for a train of a certain length, moving at a constant speed, to completely cross a bridge of another length. To solve this, we need to understand what "crossing a bridge" means in terms of distance and use the relationship between distance, speed, and time.
Calculating the Total Distance Covered
When a train crosses a bridge, it must travel a distance equal to the length of the bridge plus its own length. Think about the moment the front of the train enters the bridge and the moment the rear of the train leaves the bridge. For the entire train to be off the bridge, the front must have traveled the length of the bridge, and then the rest of the train, up to its rear, must also clear the bridge. This means the total distance the train's front (or any point on the train, consistently measured) travels is the sum of the bridge's length and the train's length.
- Length of the train = 110 m
- Length of the bridge = 165 m
- Total distance to be covered = Length of train + Length of bridge
- Total distance = \(110 \text{ m} + 165 \text{ m} = 275 \text{ m}\)
Converting Speed to Consistent Units
The speed of the train is given in kilometers per hour (km/hr), but the lengths are in meters (m). To calculate the time in seconds, we need to convert the speed from km/hr to meters per second (m/s). The conversion factor is \(1 \text{ km/hr} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s}\).
- Speed of the train = 132 km/hr
- Speed in m/s = \(132 \times \frac{5}{18} \text{ m/s}\)
- Let's simplify the conversion: \(132 \times \frac{5}{18} = \frac{132}{18} \times 5\)
- Both 132 and 18 are divisible by 6: \(132 \div 6 = 22\) and \(18 \div 6 = 3\)
- So, Speed in m/s = \( \frac{22}{3} \times 5 = \frac{110}{3} \text{ m/s}\)
Calculating the Time Required
Now that we have the total distance the train must cover and its speed in consistent units (meters and meters per second), we can use the formula:
\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)
- Total distance = 275 m
- Speed = \(\frac{110}{3} \text{ m/s}\)
- Time = \(\frac{275 \text{ m}}{\frac{110}{3} \text{ m/s}}\)
- Time = \(275 \times \frac{3}{110} \text{ seconds}\)
- Time = \(\frac{275 \times 3}{110} \text{ seconds}\)
Let's calculate the final value:
Time = \(\frac{825}{110}\) seconds
To simplify the fraction, we can divide both the numerator and the denominator by common factors. Both are divisible by 5:
\(825 \div 5 = 165\)
\(110 \div 5 = 22\)
So, Time = \(\frac{165}{22}\) seconds
Both 165 and 22 are divisible by 11:
\(165 \div 11 = 15\)
\(22 \div 11 = 2\)
So, Time = \(\frac{15}{2} \text{ seconds}\)
Time = 7.5 seconds
Therefore, the time required for the train to cross the bridge is 7.5 seconds.
| Quantity | Value | Unit |
|---|---|---|
| Train Length | 110 | m |
| Bridge Length | 165 | m |
| Total Distance | 275 | m |
| Train Speed | 132 | km/hr |
| Train Speed (converted) | \(\frac{110}{3}\) | m/s |
| Time Taken | 7.5 | seconds |
Revision Table: Train, Speed, and Distance Calculations
Here is a quick summary of the key formulas and conversions used in this problem:
- When a train crosses a stationary object of length, the total distance covered is the length of the train plus the length of the object.
- Distance = Speed × Time
- Time = Distance / Speed
- Speed = Distance / Time
- Conversion: 1 km/hr = \(\frac{5}{18}\) m/s
- Conversion: 1 m/s = \(\frac{18}{5}\) km/hr
Additional Information: Relative Speed Concepts
While this problem involved a train crossing a stationary bridge, problems often involve trains crossing other moving objects (like another train or a person). In such cases, the concept of relative speed is used. The relative speed is the difference or sum of the speeds depending on whether the objects are moving in the same or opposite directions.
- If two objects move in the same direction, their relative speed is the difference between their speeds.
- If two objects move in opposite directions, their relative speed is the sum of their speeds.
The total distance covered when two trains cross each other is always the sum of their lengths, regardless of their direction of motion. Then, this total distance is divided by the relative speed to find the time taken for them to cross each other.
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Important Questions from Problem on Trains
Eight railway stations A, B, C, D, E, F, G and H are connected either by two-way passages or one-way passages. One-way passages are from C to A, E to G, B to F, D to H, G to C, E to C and H to G. Two-way passages are between A and E, G and B, F and D, and E and D.
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:
