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Parametrics (collisions/intersections)

  1. Oct 17, 2009 #1
    1. The problem statement, all variables and given/known data
    curve1: x=5t^2-3, y=9t+2

    curve2: x=8t^2-12, y=7t+4

    find points of collision or intersection, if any

    2. Relevant equations

    see above

    3. The attempt at a solution

    I set the x functions or both curves equal to each other and got t=+/- sqrt(3)
    I did the same with the y functions and got t=1.
    I guess that this shows that there is no collision, but how do I tell if they intersect or not? I graphed this on my calc, and there seems to be an intersection, but how do I solve for it?
     
  2. jcsd
  3. Oct 18, 2009 #2

    danago

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    Gold Member

    You could possibly eliminate the parameter for each curve i.e. write each path as a cartesian equation, and then then algebraically solve for where the curves intersect.
     
  4. Oct 18, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    There will be a collision if the two objects are at the same place at the same time. That is, if [itex]5t^2- 3= 8t^2-12[/itex] and [itex]9t+2= 7t+4[/itex] for some t. The first equation reduces easily to [itex]3t^2= 9[/itex] or [itex]t^2= 3[/itex]. The second reduces easily to 2t= 2 or t= 1. Since the two x values and y values are not the same at the same time, there is no collision. That is exactly what you got.

    There will be an intersection, but no collision, if the paths pass the same point at different times. Replacing the "t" in the equations for the second path by s, we get [itex]5t^2- 3= 8s^2- 12[/itex] and [itex]9t+ 2= 7s+ 4[/itex]. That gives two equation in s and t. Can you solve those equations?
     
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