Ups and Downs on a Mountain
From her base camp on the lower edge of Mt. Baker, Polly Dent leave’s promptly at 8 a.m. on Saturday and climbs Mt. Baker, reaching its peak at exactly 4 p.m. that same day. Exhausted, she spends the night in a snow cave. She leaves the peak of Mt. Baker promptly at 8 a.m. the next day (on Sunday) and climbs down, returning to her base camp at exactly 4 p.m.
PROVE/DISPROVE: At some time on Sunday, Polly is at exactly the same height (i.e. elevation) on Mt. Baker as she was on Saturday at that same time.
Hint: Do not assume that the climber's rate is constant in either direction.
What would happen if two people replicated this experiment on the same day from the two different directions?
The hardest part of this problem is to focus on the question asked...and not on trying to find the exact time/height if they do meet.
Solution Commentary: Approach one...draw a graph, using her elevation as a function of time (after 8 am)...but the key is to use this graph to represent both trips. What do you notice?
As another approach, ask what would happen if you captured the full event on an 8-hour-long video...and then replayed both videos by projecting them on the same screen?
As a final approach that overcomes intuition-based assumptions, role-play what happens if two people face each other on opposite sides of a street at a cross-walk...and agree to start crossing at the same time (when the light turns green) and also reach their respective opposite sides of the streets at the same time (when the light turns red).
Question: Does the answer to the problem change if Polly started at the same time each day, but was allowed to finish her climb at different times (e.g. 4 pm and 2 pm respectively)?