A car moving in the morning fog overtakes a man walking at 4 km/h in the same direction. The man can see the car clearly for 3 minutes, and the maximum distance at which the car is visible ahead of him is 130 m. What is the speed of the car (in km/h)?

Introduction / Context:
This is a relative speed and visibility problem set in morning fog. A man walks along a road at 4 km/h. A car moving in the same direction overtakes him. Because of fog, the man can only see objects up to 130 m ahead. He can see the car clearly for 3 minutes. We must find the car's speed. The key idea is to relate visibility distance and the time for which the car remains within that visible range while using relative speed between the car and the man.

Given Data / Assumptions:

• Man's walking speed = 4 km/h.
• Car moves in the same direction as the man.
• Maximum distance at which the car is visible ahead of the man = 130 m.
• The man can see the car clearly for 3 minutes.
• We assume that the car becomes visible when it is next to the man and disappears when it is 130 m ahead.

Concept / Approach:
In this interpretation, the car remains visible from the moment it is level with the man until it is 130 m ahead. The relative distance covered between car and man during this time is therefore 130 m. The relative speed is the difference between their speeds because they are moving in the same direction. We find the relative speed from distance and time, and then add the man's speed to get the car's speed.

Step-by-Step Solution:
Step 1: Express visibility distance in kilometres. Visibility range ahead = 130 m = 130 / 1000 km = 0.13 km. Step 2: Express the visible time in hours. Time for which the car is visible = 3 minutes. 3 minutes = 3 / 60 hours = 0.05 hours. Step 3: Compute the relative speed. relative speed = distance / time = 0.13 / 0.05 km/h. 0.13 / 0.05 = 2.6 km/h. So the car is moving 2.6 km/h faster than the man. Step 4: Find the car's speed. Man's speed = 4 km/h. Car's speed = man's speed + relative speed = 4 + 2.6 = 6.6 km/h. Therefore, the car's speed is 6.6 km/h.
Verification / Alternative check:
Check by recomputing the relative distance using the discovered car speed. If the car moves at 6.6 km/h and the man walks at 4 km/h, their relative speed is 2.6 km/h. Over 3 minutes (0.05 hours), the relative distance covered is:
relative distance = 2.6 * 0.05 = 0.13 km = 130 m. That matches the visibility distance ahead at which the car disappears from view. Hence, 6.6 km/h is consistent with the problem data.

Why Other Options Are Wrong:
10 km/h or 8.9 km/h would give a higher relative speed and thus a larger visible distance or a shorter visible time than 130 m in 3 minutes.
7.5 km/h also produces a relative speed greater than 2.6 km/h and does not match the 130 m figure when recomputed.
5.2 km/h would give a relative speed of only 1.2 km/h, which is too small and would not carry the car far enough ahead in the given time.
Only 6.6 km/h gives exactly 130 m of relative advance in 3 minutes.

Common Pitfalls:
A key misunderstanding arises if one assumes the car is visible from 130 m behind the man to 130 m ahead, making the relative distance 260 m. That interpretation yields a different speed that does not match any of the given options. Also, students sometimes add the speeds instead of subtracting them, which is incorrect for same-direction motion. Another frequent mistake is mishandling the time conversion from minutes to hours. Carefully choosing a consistent interpretation of visibility and checking with the numbers avoids these issues.

Final Answer:
The speed of the car is 6.6 km/h.