Question

I started climbing up the hill at 6 a.m. and reached the top of the temple at 6 p.m. Next day I started coming down at 6 a.m. and reached the foothill at 6 p.m. I walked on the same road. The road is so short that only one person can walk on it. Although I varied my pace on my way, I never stopped on my way. Then which of the following must be true?

• A. My average speed downhill was greater than that uphill
• B. At noon, I was at the same spot on both the days.
• C. There must be a point where I reached at the same time on both the days.
• D. There cannot be a spot where I reached at the same time on both the days.

Hint:

In problems involving motion along a single path, if the paths are continuous and the times are fixed, there is always a point where both paths meet.

Answer 1

The Correct Option is C

The problem essentially asks about the intersection of two paths: one while going uphill and one while coming downhill. Because the road is narrow and the time taken to travel is fixed, there must be a point on the road where both paths cross at the same time. This is a classic example of the "intermediate value theorem" applied to paths. Even though the pace may vary, there must be a point on the road where both paths coincide at the same time.