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Parametrics - intersection and collision

  1. Oct 28, 2009 #1
    1. The problem statement, all variables and given/known data
    x1t= 2-cos(pi*t)
    y1t= 3+7sin(pi*t)

    x2t= 3t+2
    y2t= -(7/15)(3t+1)2 + 157/15

    Find points of intersection and collision



    2. Relevant equations
    above?


    3. The attempt at a solution

    Well, to find the intersection I think I need to eliminate the parameters for both equations so I get an equations in terms of x and y, and then set them eqaul to each other.

    x=2-3cos(pit)
    3cos(pit)=2-x
    cos(pit)=(2-x)/3

    y=3+7sin(pit)
    7sin(pit)=y-3
    sin(pit)=(y-3)/7

    using the rule cos^2+sin^2=1 and substituting, we get

    ((y-3)/7)^2 + ((2-x)/3)^2 = 1

    I guess we should solve for y here?

    Seems a little unwieldy to simplify; is there a trick or something?

    Anyways, I used the solve function on my calc and got

    y= (7*sqrt(-x^2 + 4x+5) +9) / 3


    now on to the other curve

    x=3t+2
    3t=x-2
    t=(x-2)/3

    plug the equations for t into the y equation

    I really don't know how to deal with this manually. the y2t function is complicated enough by itself, and now i have to replace t with a fraction.

    I did however graph the function and i think maybe the answers are t=1 which is (5,3) and t=1/2 which is (2, 10)
    not entirely sure.

    and then for collision, we set the x functions to be the same?

    2-3cos(pit)=3t+2
    3cos(pit)=3t
    cos(pit)=t
    I used a calc and got t= -1, -0.7898, 0.377
    can the last two values be represented in fractional from with pi?

    now i do the same thing w/ y functions

    3+7sin(pit)= -(7/15)(3t+1)2 + 157/15

    do i have to pretty much use a calculator for this?

    anyways, i got t=-1.208, 0.307, 1, 1.52

    none of the t values are the same.

    in desperate need of help plz >.>
     
  2. jcsd
  3. Oct 28, 2009 #2
    anyone? =(

    even partial help would be appreciated
     
  4. Oct 28, 2009 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Ok, I'll give you partial help. You probably meant "x1t= 2-3*cos(pi*t)" in your problem statement, right? Since that's what you used in trying to solve it. I think you are taking the correct approach. (2,10) is an intersection. I don't think it's a collision, can you tell me why? There's another real intersection as well, which I'm pretty sure is not a collision. You are pretty good at this, just keep working on it and I'll check in tomorrow. Sorry, it's really late here.
     
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