Harry, the high school teacher, lost his marbles and became the world’s best-educated hobo. He was hiking briskly across a high railroad overpass on a sunny autumn morning, when he heard a train approaching in the distance behind him. Noting that he was 3/7 of the way across the bridge, he ran ahead to the far end and hopped off just as the train caught up with him. Later, when he had time to think about it, he wondered whether he could have run back, instead, toward the train and still gotten off in time. Analyze this. Under what circumstances could he have escaped the train by running either way?
d = | r | t | |
what happened | |||
H | 4b/7 | r_{h} | t_{1} |
T | x + b | r_{t} | t_{1} |
what he thought about | |||
H | 3b/7 | r_{h} | t_{2} |
T | x | r_{t} | t_{2} |
So the train goes 7 times as fast as Harry. When he runs 4b/7, the train goes 4b, which tells us x = 3b. If the train is 3 times the length of the bridge away (or more), then Harry can run in either direction and escape catastrophe.
Query: Generalize this? What if he’s c/d of the way across instead of 3/7…?