Let's work through the problem step by step.

Given Data:

1. Faster train length = 160 meters
2. Faster train crosses a lamp post in 12 seconds.
3. Two trains cross each other in 14 seconds.
4. Speed difference between the faster train and the other train is 6 km/h.

Let the speed of the faster train be $S_f$ m/s, and the speed of the other train be $S_s$ m/s.

Step 1: Find the speed of the faster train in m/s

The speed of a train crossing a stationary object (like a lamp post) is given by the formula: $S_f = \frac{\text{Length of faster train}}{\text{Time to cross the lamp post}}$ $S_f = \frac{160}{12} = 13.33 \, \text{m/s}$

Step 2: Convert speed difference to m/s

Since the difference in speed is given in km/h, convert it to m/s: $6 \, \text{km/h} = \frac{6 \times 1000}{3600} = 1.67 \, \text{m/s}$

Thus, the speed of the other train (slower train) is: $S_s = S_f - 1.67 = 13.33 - 1.67 = 11.66 \, \text{m/s}$

Step 3: Relative speed when trains cross each other

When two trains move in opposite directions, their relative speed is the sum of their individual speeds: $\text{Relative speed} = S_f + S_s = 13.33 + 11.66 = 24.99 \, \text{m/s}$

Step 4: Calculate the length of the slower train

The total distance covered when the two trains cross each other is the sum of their lengths. Let the length of the slower train be $L_s$. The total distance covered is $L_f + L_s$.

We know that: $\text{Time to cross each other} = 14 \, \text{seconds}$ So, $\text{Total distance} = \text{Relative speed} \times \text{Time}$ $160 + L_s = 24.99 \times 14 = 349.86 \, \text{meters}$ Thus, $L_s = 349.86 - 160 = 189.86 \, \text{meters}$

Step 5: Round to the nearest integer

The length of the slower train is approximately 190 meters.

Answer:

The correct option is 3) 190 meters.

Would you like further details or explanations? Here are five related questions you could explore:

1. How would the solution change if both trains were moving in the same direction?
2. Can we solve the problem using relative speed in km/h directly?
3. How do we interpret the concept of relative speed when objects are moving in opposite directions?
4. How is the speed of a train related to the time it takes to cross a stationary object?
5. What if the speed difference between the trains were greater, say 10 km/h? How would it impact the solution?

Tip: Always convert speeds to consistent units (m/s or km/h) when solving problems involving time and distance.