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Projectile motion problem

  1. Oct 2, 2013 #1
    1. The problem statement, all variables and given/known data
    Points P and Q are in the same horizontal plane at a distance d metres apart. A projectile is shot from P in the vertical plane through PQ towards Q with a speed v m/s at an angle θ. Simultaneously another particle is shot from Q in the same vertical plane towards P with a speed of 2m/s at an angle of elevation α .

    Find an expression for the distance between the two particles when one is vertically above the other one and hence prove that they collide if [itex] sin \alpha = \frac{1}{2} sin\theta.[/itex]

    If [itex]\theta= \frac{1}{3}\pi [/itex] prove that they collide at a point above the level of PQ if...
    [tex]v^2 > \frac{gd(\sqrt{13}-1)}{6\sqrt{3}}[/tex]



    2. Relevant equations

    equations of motion

    3. The attempt at a solution

    I just don't know how to approach this one at all. I don't want to just be given the answer, just some advice or help in how to approach the problem and any assumptions would be very very much appreciated.
     
    Last edited: Oct 2, 2013
  2. jcsd
  3. Oct 2, 2013 #2

    Mentz114

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    I cannot understand the setup. Is PQ the line joining P and Q ? Also, if P and Q are in a horizontal plane, how can P or Q be vertically above the other ?
     
  4. Oct 2, 2013 #3
    I think that's what it means when it says PQ - being the line/distance joining the two. And, sorry about the second bit that was my fault copying the question down; P and Q are points on the plane, its the particles that are fired from P and Q that are/will be above each other.

    Thanks
     
  5. Oct 2, 2013 #4

    Mentz114

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    OK. You can find the the time when the projectiles are above each other by solving xP=xQ where

    xP= vt cos(θ) and xQ= d-vt cos(α)

    Does this help ?

    [edit] I made a mistake, I've corrected it.
     
  6. Oct 2, 2013 #5
    Thanks, isn't the x cordinate of a projectile in this situation generally speed*t as in [itex]v\cos(\theta) t[/itex] though?

    If i set them both to equal each other though I would get stuck as there is nothing I can do to manipulate it, is there?

    [tex]v\cos(\theta)=d-v\cos(\alpha)[/tex]
     
  7. Oct 2, 2013 #6

    Mentz114

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    I'm sorry you got caught by my mistake. I've edited my post so you can find the time now, and plug it into the equations for y to get the heights.

    You get the time from ##vt\cos(\theta)=d-vt\cos(\alpha)##
     
  8. Oct 2, 2013 #7
    OK thank you.

    I am stuck with the algebra side of it then I think as I cannot see how to extract t from that. Plus not knowing either of the angles is confusing the hell out of me, I suppose if t is found then the alpha angle could be found as it states the speed for that one (2m/s).
     
  9. Oct 2, 2013 #8

    Mentz114

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    The value of t when the projectiles are at the same x is

    ##vt\cos(\theta)=d-2t\cos(\alpha)\ \ \rightarrow\ vt\cos(\theta)+2t\cos(\alpha)=d\ \ \rightarrow t_s=\frac{d}{v\cos(\theta)+2\cos(\alpha)}##

    The y values are

    ##y_P=vt\sin(\theta),\ y_Q=2t\sin(\alpha)## put ##t_s## into these and then subtract them.
    corrected in my next post

    You need to do the algebra.
     
    Last edited: Oct 2, 2013
  10. Oct 2, 2013 #9
    Thanks for your help. :)

    With the y values, doesn't gravity need to be accounted for?
     
  11. Oct 2, 2013 #10

    Mentz114

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    Yes. Sorry I'm making a lot of mistakes. The y equations (assuming that the initial height is 0) are

    ##y_P=\int(v\sin(\theta)-gt) dt=vt\sin(\theta)-(1/2)gt^2## and ##y_Q=\int(2\sin(\alpha)-gt)dt=2t\sin(\alpha)-(1/2)gt^2##

    So the difference in height is ##vt\sin(\theta)-2t\sin(\alpha)## and the Y-values are equal (for t>0) only if ##v\sin\left( \theta\right) \,=2\,\sin\left( \alpha\right)## which is *not* what is given for this condition.
     
    Last edited: Oct 2, 2013
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