Train and Platform Length Calculation
This problem involves calculating the length of a train and a platform using concepts of speed, time, and distance, including relative speed.
Convert Speeds to Meters per Second (m/s)
First, convert the given speeds from kilometers per hour (km/h) to meters per second (m/s) because the time is given in seconds. The conversion factor is $1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}$.
- Train speed: $54 \, \text{km/h} = 54 \times \frac{5}{18} \, \text{m/s} = 3 \times 5 \, \text{m/s} = 15 \, \text{m/s}$
- Man's speed: $9 \, \text{km/h} = 9 \times \frac{5}{18} \, \text{m/s} = \frac{1}{2} \times 5 \, \text{m/s} = 2.5 \, \text{m/s}$
Calculate Train Length
When a train passes a man running in the same direction, the relative speed is the difference between their speeds. The distance the train covers relative to the man is its own length ($L_{\text{train}}$).
- Relative speed = Train speed - Man's speed $ \text{Relative Speed} = 15 \, \text{m/s} - 2.5 \, \text{m/s} = 12.5 \, \text{m/s} $
- The time taken to pass the man is 14 seconds.
- Distance = Speed $\times$ Time $ L_{\text{train}} = 12.5 \, \text{m/s} \times 14 \, \text{s} $ $ L_{\text{train}} = 175 \, \text{m} $
The length of the train is 175 meters.
Calculate Platform Length
When a train passes a platform, the total distance covered is the sum of the train's length ($L_{\text{train}}$) and the platform's length ($L_{\text{platform}}$). The train travels at its own constant speed.
- Train speed = $15 \, \text{m/s}$
- Time taken to pass the platform is 25 seconds.
- Total Distance = Train speed $\times$ Time $ L_{\text{train}} + L_{\text{platform}} = 15 \, \text{m/s} \times 25 \, \text{s} $ $ L_{\text{train}} + L_{\text{platform}} = 375 \, \text{m} $
- Substitute the calculated train length ($L_{\text{train}} = 175 \, \text{m}$): $ 175 \, \text{m} + L_{\text{platform}} = 375 \, \text{m} $ $ L_{\text{platform}} = 375 \, \text{m} - 175 \, \text{m} $ $ L_{\text{platform}} = 200 \, \text{m} $
The length of the platform is 200 meters.
Final Answer
The length of the train is 175 meters and the length of the platform is 200 meters.
Important Questions from Speed Distance and Time
- A car is moving on a horizontal surface in a straight line with a constant velocity of 3 m/s. A ball is thrown vertically upwards at time $t = 0$ from the top of the moving car with a velocity of 20 m/s. The acceleration due to gravity is 10 m/s$^2$.
At what value(s) of time $t$ in second(s), the ball is at a height of 15 m from the top of the moving car? - Velocity of an object fired directly in upward direction is given by $V = 80 – 32 \ t$, where $t$ (time) is in seconds. When will the velocity be between 32 m/sec and 64 m/sec?
In a 500 m race, P and Q have speeds in the ratio of 3: 4. Q starts the race when P has already covered 140 m.
What is the distance between P and Q (in m) when P wins the race?- An automobile travels from city A to city B and returns to city A by the same route. The speed of the vehicle during the onward and return journeys were constant at 60 km/h and 90 km/h, respectively. What is the average speed in km/h for the entire journey?
Two cars start at the same time from the same location and go in the same direction. The speed of the first car is 50 km/h and the speed of the second car is 60 km/h. The number of hours it takes for the distance between the two cars to be 20 km is ___________.
