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Math Help - relative velocities

  1. #1
    Junior Member winterwyrm's Avatar
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    relative velocities

    A 136-m-wide river flows due east at a uniform speed of 9 m/s. A boat with a speed of 1 m/s relative to the water leaves the south bank pointed in a direction 16° west of north. How long does the boat take to cross the river? Assume an xy coordinate system with the positive direction of the x axis due east and the positive direction of the y axis due north.

    Thanks in advance.
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  2. #2
    MHF Contributor
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    Quote Originally Posted by winterwyrm View Post
    A 136-m-wide river flows due east at a uniform speed of 9 m/s. A boat with a speed of 1 m/s relative to the water leaves the south bank pointed in a direction 16° west of north. How long does the boat take to cross the river? Assume an xy coordinate system with the positive direction of the x axis due east and the positive direction of the y axis due north.

    Thanks in advance.
    The current is due East.
    The boat is due N 16deg W.
    So the boat is going against the current in some way.

    The boat's speed is 1 m/sec relative to the speed of the current. In order for the boat to move to its direction, the due-west component of the boat's speed must be 1 m/sec relative to the current.
    So if the boat were to travel in a straight line, the relative velocity, V, of the boat along that straight line is
    V*cos(90 -16 deg) = 1
    V = 1 / cos(74 deg) = 3.62796 m/sec, due N 16deg W.

    The straight crossing line, d, is
    d*sin(74deg) = 136
    d = 136 / sin(74deg) = 141.48072 meters

    d is also,
    d = 3.62796*t
    where t is in seconds

    So,
    3.62796(t) = 141.48072
    t = 141.48072 / 3.62796
    t = 39 seconds ------the time the boat will cross the river.
    Last edited by ticbol; September 17th 2008 at 04:12 PM. Reason: it was N 16deg W.....not N 26deg W.
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