Two people, A and B, are 260 km apart at 9:00 a.m. A immediately starts travelling towards B with a speed of 25 km/h. At 11:00 a.m., B starts travelling towards A with a speed of 10 km/h. At what time in the afternoon (in p.m.) will they meet each other?

Introduction / Context:
This question belongs to the topic of relative speed and motion in one dimension. Two people start from different points and move towards each other at different times and speeds. You need to find the time at which they meet. Such problems require you to account for unequal start times, calculate distances covered by each person, and use the concept of relative speed once both are moving simultaneously.

Given Data / Assumptions:

• Initial distance between A and B at 9:00 a.m. = 260 km.

• A starts moving towards B at 9:00 a.m. with a speed of 25 km/h.

• B starts moving towards A at 11:00 a.m. with a speed of 10 km/h.

• Both travel along the same straight line towards each other.

• We need to find the meeting time in the afternoon (p.m.).

Concept / Approach:
First, compute how far A travels before B even starts, that is, from 9:00 a.m. to 11:00 a.m. Then, once B starts at 11:00 a.m., both are moving towards each other, so we can consider their relative speed which is the sum of their individual speeds. Using the reduced distance between them at 11:00 a.m. and their combined speed, we can calculate how long it takes for them to meet and then convert that into clock time.

Step-by-Step Solution:
Step 1: Time from 9:00 a.m. to 11:00 a.m. = 2 hours (only A is moving during this time). Step 2: Distance covered by A in 2 hours = speed * time = 25 * 2 = 50 km. Step 3: Remaining distance between A and B at 11:00 a.m. = 260 − 50 = 210 km. Step 4: From 11:00 a.m., both A and B start moving towards each other. Step 5: Relative speed when moving towards each other = 25 km/h + 10 km/h = 35 km/h. Step 6: Time taken to meet after 11:00 a.m. = remaining distance / relative speed = 210 / 35 hours. Step 7: 210 / 35 = 6 hours. Step 8: Therefore, they meet 6 hours after 11:00 a.m., which is at 5:00 p.m.

Verification / Alternative check:
We can verify by checking the distances each travels by 5:00 p.m. A travels from 9:00 a.m. to 5:00 p.m., which is 8 hours. Distance by A = 25 * 8 = 200 km. B travels from 11:00 a.m. to 5:00 p.m., which is 6 hours. Distance by B = 10 * 6 = 60 km. Total distance covered together = 200 + 60 = 260 km, which matches the original distance between them. Thus, they indeed meet at 5:00 p.m.

Why Other Options Are Wrong:
• 6:00 p.m.: If they met at 6:00 p.m., A would have travelled 9 hours (225 km) and B 7 hours (70 km), totaling 295 km, which exceeds the initial 260 km.
• 6:30 p.m.: This gives even more total distance, making it even more impossible for them to meet at that time if they started 260 km apart.
• 7:00 p.m.: The total distance covered by then would be much larger than 260 km, so meeting earlier must have already occurred at 5:00 p.m.

Common Pitfalls:
A common mistake is to assume both start at the same time and directly use relative speed for the entire distance, ignoring the delay in B's start. Another error is miscounting the elapsed time or miscomputing the remaining distance after A's initial solo movement. Carefully separating the motion into two phases (before and after B starts) and then applying relative speed in the second phase avoids these issues.

Final Answer:
A and B will meet each other at 5:00 p.m..