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Algebra II tossed up with Physics

  1. Jun 13, 2003 #1
    Two motorized toy boats (they are 5 meters apart)
    are released in a pool at time t = 0. Boat 1 travels 35 degree North of east at a rate of 0.65 meter per second. Boat 2 travels due east at a rate of 0.4 meter per second. Good grief, I forgot the size of the pool, can we say it is AxB or assign any number such as 10m X 20m?

    > a. Write a set of parametric equations to describe the path of each boat.
    > b. At what point will each boat hit the east edge of the pool?
    > c. At what point do the paths of the two boats cross?
  2. jcsd
  3. Jun 14, 2003 #2
    First of all, welcome to the forums.
    In the homework help forum you are asked to show us where you got stuck in order to help you.
    But anyway, your question needs a lot more info to be solved (at least this is what i see).
    First, we need to know more about the "5 meters apart" fact, you need to know in which direction (relative to each other) the boats are places initially.
    Second, you need to give coordination points of the boats (or at least a clue about that) initially relatively to any of the edge of the pool, otherwise we cannot know when it will hit the pool (you see, if the boats started to move at the middle of the pool, they will not hit the east edge like if they started at the west edge of the pool, i think this is obvious).
    I am sure you will be helped as soon as u provide the information, and where you got stuck in the question.
  4. Jun 14, 2003 #3
    a. Write a set of parametric equations to describe the path of
    each boat.
    Boat 1 has subscript 1 boat 2 has subscript 2
    each boat has x and y coordinates which are a function of t
    boat 1 has x1(t) and y1(t) boat 2 has x2(t) and y2(t)
    x1(t) = 0.65 cos(35)t y1(t) = 0.65 sin(35)t x2(t) =
    y2(t) = constant
    This is what my good neighbor told me. Since he was so kind and prompt, I didn't have a heart to ask him where those sin and cos coming from.

    Boat 1's coordnate (0,0)
    Boat 2's coordinate (0,5)

    The width of pool is 15m. I guess the length of the pool won't be the issue.
  5. Jun 15, 2003 #4
    Ok, this is better.
    First of all, to clear out what where the cos() and sin() comes from.
    The velocity of each boat is a vector, vectors can be analyzed into components, in our case horizontal (x) and vertical (y) component.
    To find the magnitude of the horizontal component, multiply the magnitude of the vector by Cos() of the angle between the vector and the x-axis (you can figure this out if you draw a triangle with a side having the length of the vector, and another side on the X axis, and remember the definition of Cos() ).
    So, the magnitude the horizontal component of the velocity of the fisrt boat (Vx1) = V1*cos(35) .
    The magnitude of the vertical component of the velocity is the velocity of the boat multiplied by the Sin() of the angle between the velocity and the X-axis (which is equal to the Cos() of the angle between the velocity and the Y-axis).
    So, Vy1 = V1*Sin(35) .
    And it is well known that
    S = V*t
    (where S is the displacement)
    So :
    x1 = Vx1*t = V1*Cos(35)*t
    y1 = Vy1*t = V1*Sin(35)*t

    And do the same for the other boat. (remember that the angle between the velocity of boat2 and X-Axis is 0).

    For question B, boat1 will hit the edge when x1(t) = the width of the pool, and boat2 will hit it when x2(t2) = the width of the pool.

    Now to solve question C, let's change the parametric equations a little so that they refer to the origin point. (add the value of x1 when t=0)

    x1(t1) = V1*Cos(35)*t2 + 0
    y1(t1) = V1*Sin(35)*t2 + 0
    x2(t2) = V2*t2 + 0
    y2(t2) = 0 + 5

    Now, when the boats paths meet x1(t1)=x2(t2) and y1(t1)=y2(t2) (or in other words, we are trying to find the point that both pathes share).
    Note that the time in which each boat will reach this point might not be the same ! (this is why i called them t1 and t2).
    Solve those two equations, and you will find the value of t1 or t2, use that to find x and y (from the parametric equations).

    If u are stuck in any certain thing, please ask.
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