Two men start from the same point P, where the first man runs from 11:00 a.m. with rest breaks, and a second man starts at 2:00 p.m. and runs non stop at 15 km/h towards him. At what time in the afternoon does the second man catch the first man?

Introduction / Context:
This problem combines uniform speed motion with a repeated pattern of running and resting. It tests your understanding of how to track positions over time when one person moves in cycles and another moves continuously. It is a typical time and distance catch up scenario seen in competitive examinations.

Given Data / Assumptions:
The first man starts running from point P at 11:00 a.m.
His running speed is 10 km/h.
He runs for 2 hours, then rests for 1 hour, and repeats this run rest cycle.
The second man starts from the same point P at 2:00 p.m. and runs non stop at 15 km/h.
Both men run in the same direction along a straight path.
We need to find the clock time in p.m. at which the second man catches the first man.

Concept / Approach:
We treat the motion piecewise. For the first man, we compute how far he has reached at different times according to the 2 hour running and 1 hour resting pattern. For the second man, the speed is constant, so his distance from P is simply speed multiplied by time after 2:00 p.m. They meet when both have covered the same distance from P. Setting these distances equal allows us to solve for the meeting time.

Step-by-Step Solution:
Step 1: From 11:00 a.m. to 1:00 p.m., the first man runs for 2 hours at 10 km/h, so distance = 10 * 2 = 20 km.Step 2: From 1:00 p.m. to 2:00 p.m., he rests, so his distance stays 20 km from P.Step 3: At 2:00 p.m., the second man starts at P. At that moment the first man is 20 km ahead.Step 4: From 2:00 p.m. to 4:00 p.m., the first man is again in a 2 hour running phase, so his speed remains 10 km/h.Step 5: Let t be the time in hours after 2:00 p.m. when the second man catches the first man, during this 2:00 to 4:00 running period.Step 6: Distance of first man from P at time t after 2:00 p.m. = 20 + 10 * t.Step 7: Distance of second man from P at the same moment = 15 * t.Step 8: At meeting, distances are equal, so 20 + 10 * t = 15 * t.Step 9: Rearranging: 20 = 5 * t, so t = 4 hours.Step 10: Add 4 hours to 2:00 p.m., we get 6:00 p.m. This seems to contradict the earlier computed pattern, so we must carefully place the meeting inside the correct segment.Step 11: Actually, we need to check the activity pattern around 4:00 p.m. The first man runs 11 to 1, rests 1 to 2, runs 2 to 4, and then rests 4 to 5.Step 12: If t = 4 hours after 2:00 p.m., that is 6:00 p.m., which lies outside the 2:00 to 4:00 running period, so our assumption that he is running during the full catch up interval is wrong.Step 13: Instead, calculate the positions at 4:00 p.m. First man has run 11 to 1, rested 1 to 2, and run 2 to 4, so total running time is 4 hours, giving distance 40 km from P.Step 14: At 4:00 p.m., second man has been running from 2:00 to 4:00, that is 2 hours at 15 km/h, so distance = 30 km.Step 15: From 4:00 to 5:00 p.m., the first man rests and stays at 40 km. The second man continues at 15 km/h.Step 16: Let u be the time in hours after 4:00 p.m. when the second man catches the first man. At that time, his distance from P is 30 + 15 * u and the first man remains at 40 km.Step 17: Set 30 + 15 * u = 40. This gives 15 * u = 10, so u = 10 / 15 = 2 / 3 hours.Step 18: 2 / 3 hours is 40 minutes. So the meeting time is 4:00 p.m. plus 40 minutes, that is 4:40 p.m.

Verification / Alternative check:
At 4:40 p.m., the time from 11:00 a.m. is 5 hours and 40 minutes, or 17 / 3 hours. The first man has run for 4 hours and rested for 1 hour and 40 minutes in total. His distance from P is 40 km. The second man has run from 2:00 p.m. to 4:40 p.m., which is 2 hours and 40 minutes or 8 / 3 hours. His distance at 15 km/h is 15 * 8 / 3 = 40 km. The positions match, confirming the solution.

Why Other Options Are Wrong:
6:20 p.m. and 6:00 p.m. would place the meeting too late, when the second man has already gone far beyond 40 km while the first man is still near his earlier position. The equations for distances no longer balance there.
5:30 p.m. is also incorrect, because at that time the second man distance minus the first man distance does not equal zero when computed using their exact running and resting pattern.
5:00 p.m. is simply a midpoint guess and does not satisfy the distance equality either.

Common Pitfalls:
Many learners try to average speeds or ignore the rest periods, which leads to incorrect times. Another common mistake is to assume a single linear equation without checking which phase of the run rest cycle the first man is in. Carefully tracking time segments and checking whether the person is running or resting at each stage is essential for solving such problems correctly.

Final Answer:
The second man catches the first man at 4:40 p.m.