One train leaves a station at 3:00 p.m. and travels in a straight line at 70 km/h. A second train starts from the same station in the same direction at 4:30 p.m. at 85 km/h. At what time will the faster train catch up with the slower train?

Introduction / Context:
This problem is a classic example of a chase or catch up scenario involving two trains leaving at different times and travelling at different speeds. Questions like this help develop intuition for relative motion in the same direction, time differences, and distance travelled before one object catches the other.

Given Data / Assumptions:

• Train 1 leaves at 3:00 p.m. with speed 70 km/h.
• Train 2 leaves the same station at 4:30 p.m. with speed 85 km/h.
• Both trains move along the same straight track in the same direction.
• We ignore stops, acceleration, and any other delays.

Concept / Approach:
First, we find how far the first train travels before the second train even starts. That distance becomes the head start. Then we use the relative speed between the two trains to calculate how long the second train takes to close that distance. Finally, we add this time to the start time of the second train to get the actual meeting time.

Step-by-Step Solution:
Time gap between departures = 4:30 p.m. − 3:00 p.m. = 1.5 hours. Distance travelled by Train 1 in 1.5 hours = 70 * 1.5 = 105 km. Relative speed when both are moving = 85 − 70 = 15 km/h. Time taken by Train 2 to make up 105 km at 15 km/h = 105 / 15 = 7 hours. Train 2 starts at 4:30 p.m., so meeting time = 4:30 p.m. + 7 hours = 11:30 p.m.

Verification / Alternative check:
At 11:30 p.m., time since 3:00 p.m. for Train 1 is 8.5 hours, so it covers 70 * 8.5 = 595 km. Time since 4:30 p.m. for Train 2 is 7 hours, so it covers 85 * 7 = 595 km. Both distances match, confirming that the trains meet at 11:30 p.m.

Why Other Options Are Wrong:
9:50 p.m., 10:10 p.m., and 10:30 p.m. all correspond to less time for the faster train to catch the slower one, meaning it would not yet have closed the 105 km gap at a relative speed of 15 km/h. Only 11:30 p.m. satisfies the time and distance equations simultaneously.

Common Pitfalls:
Students often forget to account for the delayed start of the second train and treat both as starting together. Another mistake is to add the speeds instead of taking the difference in a same direction chase problem. Finally, some learners mis-handle the conversion from fractional hours to clock times, so practising this step is important.

Final Answer:
The faster train will catch up with the slower train at 11:30 p.m.