Understanding Train Crossing Problems

This problem involves calculating the speed of a train based on the time it takes to cross a platform. To solve this, we need to determine the total distance the train travels while crossing the platform and then use the formula relating speed, distance, and time.

Calculating Total Distance

The train needs to cover its own length plus the length of the platform to completely cross it.

• Train Length ($L_{train}$): 175 m
• Platform Length ($L_{platform}$): Twice the train's length = $2 \times 175 \text{ m} = 350 \text{ m}$
• Total Distance ($D_{total}$) = Train Length + Platform Length

Using LaTeX notation:

$D_{total} = L_{train} + L_{platform}$

$D_{total} = 175 \text{ m} + 350 \text{ m}$

$D_{total} = 525 \text{ m}$

Calculating Speed in m/s

The time taken to cover this total distance is given.

• Time ($t$): 27 seconds
• Speed ($S$) is calculated using the formula: $S = \frac{\text{Distance}}{\text{Time}}$

Substituting the values:

$S = \frac{D_{total}}{t}$

$S = \frac{525 \text{ m}}{27 \text{ s}}$

Now, we simplify this fraction:

$S = \frac{525}{27} \text{ m/s}$

We can divide both numerator and denominator by 3:

$S = \frac{175}{9} \text{ m/s}$

Converting Speed to km/h

The question asks for the speed in kilometers per hour (km/h). To convert speed from meters per second (m/s) to km/h, we multiply by the conversion factor $\frac{18}{5}$.

Speed in km/h = Speed in m/s $\times \frac{18}{5}$

Let's apply this conversion:

Speed (km/h) = $\frac{175}{9} \times \frac{18}{5}$

Now, perform the multiplication and simplification:

Speed (km/h) = $\frac{175 \times 18}{9 \times 5}$

Simplify by cancelling common factors:

• Divide 18 by 9: $\frac{18}{9} = 2$
• Divide 175 by 5: $\frac{175}{5} = 35$

So the expression becomes:

Speed (km/h) = $35 \times 2$

Speed (km/h) = $70$

Final Answer

The speed of the train is 70 km/h.