From the time the front of a train enters a platform, it takes 25 seconds for the back of the train to leave the platform, while travelling at a constant speed of 54 km/h. At the same speed, it takes 14 seconds to pass a man running at 9 km/h in the same direction as the train. What is the length of the train and that of the platform in meters, respectively?
175 and 200
Train Length and Platform Length Calculation
This problem involves concepts of speed, distance, and time, specifically dealing with relative speed when a train passes an object (like a man) and when it passes a platform. We need to calculate two unknowns: the length of the train and the length of the platform.
Converting Speeds to Consistent Units
First, it's essential to convert all given speeds from kilometers per hour (km/h) to meters per second (m/s) because the time is given in seconds and we need lengths in meters. To convert km/h to m/s, we multiply by a factor of \(\frac{5}{18}\).
- Train's speed: \(54 \, \text{km/h}\)
- Man's speed: \(9 \, \text{km/h}\)
Let's convert these speeds:
- Train speed (\(V_T\)): \[V_T = 54 \, \text{km/h} \times \frac{5}{18} \, \text{m/s} = 3 \times 5 \, \text{m/s} = 15 \, \text{m/s}\]
- Man's speed (\(V_M\)): \[V_M = 9 \, \text{km/h} \times \frac{5}{18} \, \text{m/s} = 0.5 \times 5 \, \text{m/s} = 2.5 \, \text{m/s}\]
| Entity | Speed (km/h) | Speed (m/s) |
|---|---|---|
| Train | 54 | 15 |
| Man | 9 | 2.5 |
Calculating the Length of the Train
When a train passes a man running in the same direction, the effective speed is the relative speed, which is the difference between the train's speed and the man's speed. The distance covered by the train to completely pass the man is equal to the length of the train itself.
- Time taken to pass the man = 14 seconds
- Man is running in the same direction as the train.
Relative speed (\(V_{\text{relative}}\)):
\[V_{\text{relative}} = V_T - V_M = 15 \, \text{m/s} - 2.5 \, \text{m/s} = 12.5 \, \text{m/s}\]Now, we can find the length of the train (\(L_T\)) using the formula: \(\text{Distance} = \text{Speed} \times \text{Time}\).
\[L_T = V_{\text{relative}} \times \text{Time taken to pass man}\] \[L_T = 12.5 \, \text{m/s} \times 14 \, \text{s}\] \[L_T = 175 \, \text{m}\]So, the length of the train is 175 meters.
Calculating the Length of the Platform
When a train passes a platform, the total distance covered by the train is the sum of its own length and the length of the platform. The train's constant speed is used for this calculation.
- Time taken to cross the platform = 25 seconds
- Train's speed (\(V_T\)) = 15 m/s
- Length of train (\(L_T\)) = 175 m (calculated above)
The total distance covered is \(L_T + L_P\), where \(L_P\) is the length of the platform.
\[L_T + L_P = V_T \times \text{Time taken to cross platform}\] \[175 \, \text{m} + L_P = 15 \, \text{m/s} \times 25 \, \text{s}\]First, calculate the total distance:
\[15 \, \text{m/s} \times 25 \, \text{s} = 375 \, \text{m}\]Now, substitute this back into the equation:
\[175 \, \text{m} + L_P = 375 \, \text{m}\]To find \(L_P\), subtract the length of the train from the total distance:
\[L_P = 375 \, \text{m} - 175 \, \text{m}\] \[L_P = 200 \, \text{m}\]So, the length of the platform is 200 meters.
Summary of Lengths
Based on our calculations:
- Length of the train = 175 meters
- Length of the platform = 200 meters
Therefore, the lengths of the train and the platform are 175 meters and 200 meters, respectively.
Important Questions from Numerical Computation
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What is the value of \(1 + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + \frac{1}{{256}} + \ldots ?\)
A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?
