This problem involves calculating the speed of trains moving in opposite directions. When two objects move towards each other or in opposite directions, their relative speed is the sum of their individual speeds. The total distance covered when two trains cross each other is the sum of their lengths.

Calculating Total Distance and Relative Speed of Trains

First, let's identify the given information:

• Length of the first train ($L_1$) = 600 meters
• Length of the second train ($L_2$) = 1000 meters
• Time taken for the trains to cross each other ($T$) = 96 seconds
• Speed of the first train = Speed of the second train (let this speed be $v$)
• The trains are moving in opposite directions.

When the two trains cross each other, the total distance they cover relative to each other is the sum of their lengths.

Total Distance ($D$) = Length of first train + Length of second train

$$D = L_1 + L_2$$

$$D = 600 \text{ m} + 1000 \text{ m} = 1600 \text{ m}$$

Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds.

Let the speed of each train be $v$ meters per second (m/s).

Relative Speed ($V_{relative}$) = Speed of first train + Speed of second train

$$V_{relative} = v + v = 2v \text{ m/s}$$

Applying the Speed, Distance, Time Relationship

We know the relationship between speed, distance, and time:

Distance = Speed × Time

So, Total Distance = Relative Speed × Time Taken

$$D = V_{relative} \times T$$

Substitute the values we have:

$$1600 \text{ m} = (2v \text{ m/s}) \times 96 \text{ s}$$

Now, we can solve for $v$:

$$2v = \frac{1600}{96}$$

$$v = \frac{1600}{96 \times 2}$$

$$v = \frac{1600}{192}$$

Let's simplify the fraction:

Divide both numerator and denominator by common factors. Both are divisible by 16:

$$v = \frac{1600 \div 16}{192 \div 16} = \frac{100}{12}$$

Both are divisible by 4:

$$v = \frac{100 \div 4}{12 \div 4} = \frac{25}{3} \text{ m/s}$$

Converting Speed from m/s to km/h

The speed we found is in meters per second (m/s). The options are given in kilometers per hour (km/h). To convert speed from m/s to km/h, we multiply by $\frac{18}{5}$.

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

$$v \text{ in km/h} = \frac{25}{3} \times \frac{18}{5}$$

We can simplify this expression:

$$v \text{ in km/h} = \frac{25}{5} \times \frac{18}{3}$$

$$v \text{ in km/h} = 5 \times 6$$

$$v \text{ in km/h} = 30 \text{ km/h}$$

So, the speed of each train is 30 km/h.

Revision Table: Key Concepts

Additional Information on Relative Speed Problems

Understanding relative speed is crucial for solving problems involving moving objects like trains, boats, or people.

• Opposite Directions: When objects move towards each other or in opposite directions, their relative speed is the sum of their individual speeds. This is because the distance between them decreases rapidly.
• Same Direction: When objects move in the same direction, their relative speed is the difference between their individual speeds (faster speed minus slower speed). This is because the distance between them changes at a slower rate.
• Crossing Point/Time: The time it takes for two objects to cross each other or for one object to overtake another depends on the total distance that needs to be covered relative to each other and their relative speed. For trains, this total distance is typically the sum of their lengths when considering them fully passing each other.

Always pay attention to the units given in the problem (meters, kilometers, seconds, hours) and ensure consistency or convert them appropriately before calculation.