To solve this problem, we need to determine the time it takes for the two trains moving in opposite directions to completely cross each other.

Step-by-Step Solution

1. Convert the speeds to the same unit: The speeds are given in kilometers per hour (km/h), while the lengths are in meters. We should convert the speeds from km/h to meters per second (m/s).

   $1 \, \text{km/h} = \frac{1000 \, \text{m}}{3600 \, \text{s}} = \frac{5}{18} \, \text{m/s}$

   Thus, the speeds in meters per second:

   $\text{Speed of first train} = 60 \, \text{km/h} = 60 \times \frac{5}{18} \, \text{m/s} = \frac{300}{18} \, \text{m/s} = 16.\overline{6} \, \text{m/s}$

   $\text{Speed of second train} = 90 \, \text{km/h} = 90 \times \frac{5}{18} \, \text{m/s} = \frac{450}{18} \, \text{m/s} = 25 \, \text{m/s}$

2. Calculate the relative speed: When two objects move in opposite directions, their relative speed is the sum of their speeds.

   $\text{Relative speed} = 16.\overline{6} \, \text{m/s} + 25 \, \text{m/s} = 41.\overline{6} \, \text{m/s}$

3. Calculate the total distance to be covered: The total distance covered when the two trains completely cross each other is the sum of their lengths.

   $\text{Total distance} = 200 \, \text{m} + 300 \, \text{m} = 500 \, \text{m}$

4. Determine the time taken to cross each other: Time taken to cross each other is given by the formula:

   $\text{Time} = \frac{\text{Total distance}}{\text{Relative speed}}$

   Substituting the values:

   $\text{Time} = \frac{500 \, \text{m}}{41.\overline{6} \, \text{m/s}} \approx \frac{500}{41.67} \, \text{s} \approx 12 \, \text{s}$

Conclusion

The two trains will completely cross each other in approximately 12 seconds.

Would you like to know more details about this solution, or do you have any questions?

Here are 5 related questions to consider:

1. What would happen to the time if the trains were moving in the same direction?
2. How would the time change if the lengths of the trains were doubled?
3. What is the impact on crossing time if the speed of one train is increased by 50%?
4. How can we calculate the crossing time if there are more than two trains?
5. How is relative speed used in other types of problems?

Tip: Always ensure that units are consistent when solving physics or math problems to avoid mistakes.