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Math Help - Vectors help

  1. #1
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    Unhappy Vectors help

    Boat A has initial position (-8,9) and has a velocity vector [4,-1].
    Boat B has initial position (1.-9).

    Boat B has a maximum speed of 8km/h. Determine the time and velocity vector [a,b] for the collision to occur as soon as possible.

    So hard..don't know how to do it.
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  2. #2
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    Re: Vectors help

    There may be other ways.

    Step 1. Let t be the time to the collision. Using a and b, write two equations on t. Dividing one equation by the other, you cat get a linear relation between a and b: they have to satisfy this relation in order for the boats to meet.

    Step 2. Allowed points (a, b) are inside the circle with center (0, 0) and radius 8. The circle cuts a segment from the line obtained in step 1. Each point on the segment gives rise to some t; you need to find the point with the smallest t.

    Step 3. From one of the equations from step 1 express t through a and draw a graph of t(a). Find the minimum of t when a belongs to the segment found in step 2.
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  3. #3
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    Re: Vectors help

    Sorry, I still don't understand. I'm pretty dumb.
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  4. #4
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    Re: Vectors help

    Let's take it one step at a time.
    Quote Originally Posted by emakarov View Post
    Let t be the time to the collision. Using a and b, write two equations on t.
    If you can do this, write the first thing in the outline from post #2 that you are having trouble with.
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  5. #5
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    Re: Vectors help

    What do you mean "write two equations on t"?
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  6. #6
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    Re: Vectors help

    The values a, b and t are related by the fact that there is an encounter in t hours. In the horizontal direction, boat B, which moves right at a km/h, starts 1 - (-8) = 9 km to the right of boat A, which moves right at 4 km/h. The fact that they meet in t hours gives an equation that uses a and t. Similarly, there is an equation for the vertical direction.
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  7. #7
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    Re: Vectors help

    Quote Originally Posted by emakarov View Post
    The fact that they meet in t hours gives an equation that uses a and t.
    Hmm...I don't know how to construct the equation.
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  8. #8
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    Re: Vectors help

    The relative speed at which A catches up with B is 4 - a. There are 9 km to catch up in t hours.

    Alternatively, boat A will be at x-coordinate -8 + 4t in t hours. Similarly, boat B will be at 1 + at. These coordinates coincide when they meet.
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  9. #9
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    Re: Vectors help

    Quote Originally Posted by emakarov View Post
    The relative speed at which A catches up with B is 4 - a. There are 9 km to catch up in t hours.

    Alternatively, boat A will be at x-coordinate -8 + 4t in t hours. Similarly, boat B will be at 1 + at. These coordinates coincide when they meet.
    So the y-coordinate for boat A would be 9-t and for boat B would be -9+bt ?
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  10. #10
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    Re: Vectors help

    Yes.
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  11. #11
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    Re: Vectors help

    Okay, so now do I equate them?
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  12. #12
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    Re: Vectors help

    I am sure they won't blow up if you do
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  13. #13
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    Re: Vectors help

    Quote Originally Posted by emakarov View Post
    I am sure they won't blow up if you do
    I got: a=(-9/t)+4 and b=(18/t)-1
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  14. #14
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    Re: Vectors help

    My original idea was to eliminate t and to express b through a.
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  15. #15
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    Re: Vectors help

    Quote Originally Posted by emakarov View Post
    My original idea was to eliminate t and to express b through a.
    How do I eliminate t?
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