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Two students are canoeing on a river. While heading upstream, they accidentally drop an empty bottle overboard. They then continue paddling for 1.9 h , reaching a point 1.2 km farther upstream. At this point they realize that the bottle is missing and, driven by ecological awareness, they turn around and head downstream. They catch up with and retrieve the bottle (which has been moving along with the current) 4.6 km downstream from the turn-around point.

How do I draw a diagram for a case like this, where I have to think about both the x and y components of speed, position, along with the time?

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  • $\begingroup$ I think that whoever made up the problem intended you to idealize it as a one-dimensional problem. That's a reasonable idealization if we assume the width of the river is a lot less than a kilometre. $\endgroup$ – Andrew Steane Jul 19 at 21:36
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This seems to be an unidimensional problem. I guess you may assume uniform speeds for both the stream and the students relative to the stream. That should simplify the diagram. You only need to represent x, for both the bottle and the students, as a function of time. The speed of the students will be the speed of the stream plus or minus their velocity relative to the stream, depending on their direction, and the speed of the bottle is the speed of the stream. The points at which both graphs intersect are when the bottle is dropped and retrieved. If you want something more general, this may also be a good starting point because it will give you insight for what should happen.

Ps: For me makes more sense to put the time in the "y"-direction because it is easier to understand where they are on the river.

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  • $\begingroup$ Wait.. why do we need only the x direction for the students and the bottle? In order for the students to pick up the bottle, don't the y coordinates position of the bottle and students have to match as well? $\endgroup$ – John Oct 10 '17 at 3:24
  • $\begingroup$ Am I severely misinterpreting the problem? $\endgroup$ – John Oct 10 '17 at 4:05
  • $\begingroup$ Yes and no. Yes you need to know both the x, y and z. And no, because although the river lives in a 3-dimensional world you can see it as an 1-dimensional object. Think about a thread that is straight and on which you made some marks. The distance between these marks (measure along the thread) wont change if the thread is no longer straight. That is why the distance is given like "upstream" and "downstream".That means that the students travelled 1.2 miles along the river. I should have ben clearer. The x is not really "x" but the "distance" to some point measure along the river. $\endgroup$ – Jonhy Oct 10 '17 at 4:25
  • $\begingroup$ But we don't know if the students traveled parallel to the river or not.. if they didn't, then the speed of the students = speed of the stream +- speed of students relative to stream doesn't apply because we need to consider the velocities if they don't travel parallel, giving us a vector diagram. From there, I don't really know what to do.. $\endgroup$ – John Oct 10 '17 at 4:33

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