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Thread: boat on a river, boat on a lake

  1. #1 boat on a river, boat on a lake 
    M
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    I enjoy the puzzles sometimes posted on this forum, so I'd like to share one of my own. It involves a very simple analysis, but still provides some interesting insight. I'll use a numerical example (instead of symbolical) because some people like to work with numbers rather than symbols.

    Peter lives on the shore of a large lake and drives his motorboat from his home (point A) to the other shore (point B) every day to get supplies, and then back home. Points A and B are 4 miles apart, the water of the lake is exactly quiessent (no currents) and Peter's boat has a constant speed of 16 miles per hour.

    His brother James lives on the shore of a river and drives his motorboat from his home (point C) downstream to point D every day to get supplies, and then back home. Points C (upstream) and D (downstream) are 4 miles apart. The river water has a uniform speed of 8 miles per hour, and James' boat is identical to Peter's, i.e. it makes 16 miles per hour, relative to the water.

    Both guys commute the same distance to get supplies and get back home (from A to B to A, or from C to D to C, respectively). Their boats make the same speed. However, unlike Peter, James has to deal with the river current, which works with him or against him, depending on the direction.

    The question: Whose commute takes a longer time (or are they the same)?

    Neglect secondary effects such as air resistance, the effect of the supplies' weight, assumed personalities of James and Peter, or anything else that wasn't mentioned.

    The math is really easy and I have no doubt that anyone can do it. I am more interested in the interpretation of the result. Is it intuitive/counter-intuitive?


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  3. #2  
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    Obviously the guy on the lake saves ten minutes per round trip, Provided the stream flow is uniform across it's width [which it won't be] I'll ignore the effects of that, and bernoulli :wink: and anything else that might ake effect!


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  4. #3  
    Forum Ph.D. william's Avatar
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    Wish I'd thought of this one...

    I agree with billco. The "river rat" :wink: will spend 10 more minutes on the water than the other dude.

    Quote Originally Posted by M
    The math is really easy and I have no doubt that anyone can do it. I am more interested in the interpretation of the result. Is it intuitive/counter-intuitive?
    At first this seems counter-intuitive - until I asked myself what happens if the speed of the current is greater or equal to (or even comparable to but less than) the speed of the boat....

    It's interesting that this also applies to driving on land.

    Good one!

    Cheers,
    william
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
    ~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
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  5. #4  
    M
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    I think there is something we can learn from this example. Many people (including myself) intuitively assume the traveling time is the same, because the guy on the river is traveling the same distances with the stream and against the stream, so the effect of the river might cancel out.

    Well, it's not the case. Distance doesn't matter, here, it's time that matters: Much more time is spent going against the stream than with the stream, and that's what's killing him.

    What about this problem, then: It's windy (constant wind in constant direction) and you are participating in a bike race on a circular track. Sometimes you are going with the wind, sometimes against it. What is the right strategy when going against the wind? Should you spare your energy and take it slowly (and put all your energy into the section with the wind), or should you try to get through it as fast as possible (and relax when going with the wind)?
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  6. #5  
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    THis is a far more complex problem. Firstly wind resistance is proportional to the sqaure of it's speed[if I remember correctly] so going flat out will not produce a gain. There are two types of muscular tissue, the type involved with stamina will be required here, I would recommend a constant effort [which would mean gear changes and speed variation] together with 'tucking in' behind others or cycling on the leeward side. They will all try that so you end up going in a straight line out through the stadium entrance. It's a mix clearly, of stamina and strategy as is seen in the Tour De France.

    And your previous problem I too thought [initially] that any gain in one direction would be cancelled in the other[stream direction].

    And as William correctly pointed out if the speed of the boat can only match the speed of flow then either you will never get there, or never get back!

    'River rat' can only 'win' if his river is tidal and he chooses the right time of day to start.
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