Analyzing Train Travel Times

This problem involves two trains traveling between Hyderabad and Vijayawada. We need to find the time when they meet or cross each other. Let's break down the journey for each train.

Train 1 Details

• Departure Station: Hyderabad
• Departure Time: 4:00 A.M.
• Arrival Station: Vijayawada
• Arrival Time: 8:00 A.M.
• Total Travel Time for Train 1: 8:00 A.M. - 4:00 A.M. = 4 hours.

Train 2 Details

• Departure Station: Vijayawada
• Departure Time: 6:00 A.M.
• Arrival Station: Hyderabad
• Arrival Time: 9:30 A.M.
• Total Travel Time for Train 2: 9:30 A.M. - 6:00 A.M. = 3 hours and 30 minutes = 3.5 hours = $\frac{7}{2}$ hours.

Calculating Speeds and Meeting Point

Let the distance between Hyderabad and Vijayawada be '$D$' kilometers.

• Speed of Train 1 ($S_1$) = Distance / Time = $\frac{D}{4}$ km/hr.
• Speed of Train 2 ($S_2$) = Distance / Time = $\frac{D}{7/2}$ = $\frac{2D}{7}$ km/hr.

We need a common reference time to calculate when they meet. Let's consider 6:00 A.M., which is the departure time of the second train.

• At 6:00 A.M., Train 1 has already been traveling for 2 hours (from 4:00 A.M.).
• Distance covered by Train 1 by 6:00 A.M. = Speed $\times$ Time = $\frac{D}{4} \times 2 = \frac{D}{2}$.
• This means at 6:00 A.M., Train 1 is exactly halfway between Hyderabad and Vijayawada.
• At 6:00 A.M., Train 2 is just starting from Vijayawada.
• The distance remaining between the two trains at 6:00 A.M. is $D - \frac{D}{2} = \frac{D}{2}$.

Calculating Relative Speed and Crossing Time

Since the trains are traveling towards each other, their relative speed is the sum of their individual speeds.

• Relative Speed = $S_1 + S_2 = \frac{D}{4} + \frac{2D}{7}$
• To add these fractions, we find a common denominator (28):
• Relative Speed = $\frac{7D}{28} + \frac{8D}{28} = \frac{15D}{28}$ km/hr.

Now, we can calculate the time it takes for the trains to cover the remaining distance ($\frac{D}{2}$) between them at their relative speed.

• Time to meet (after 6:00 A.M.) = Remaining Distance / Relative Speed
• Time = $\frac{D/2}{15D/28} = \frac{D}{2} \times \frac{28}{15D}$
• Time = $\frac{28}{30}$ hours = $\frac{14}{15}$ hours.

Let's convert this time into minutes:

• Time in minutes = $\frac{14}{15} \times 60$ minutes = $14 \times 4$ minutes = 56 minutes.

The trains will meet 56 minutes after 6:00 A.M.

Final Crossing Time

Crossing Time = 6:00 A.M. + 56 minutes = 6:56 A.M.