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Two particle system involving conservation laws

  1. Sep 28, 2014 #1
    1. The problem statement, all variables and given/known data

    A billiard ball of mass M is initially at rest on a horizontal frictionless table. Another ball
    of mass m < M and velocity ##\vec{v}## in the positive x-direction hits the first ball in a pefectly elastic
    collision. After the collision, the balls move with (unknown) velocities ##\vec{U} ## and ##\vec{u}## respectively
    (not necessarily in the x-direction).
    Find the maximum amount of kinetic energy ##∆T## that the second ball can impart on
    the first ball.


    2. Relevant equations
    ##T=\frac{1}{2}mv^2##
    ##\rho_i=\rho_f##
    ##T_i=T_f##
    3. The attempt at a solution
    I'm entirely stuck and I'm not sure how to tackle this problem. Any help is appreciated.
     
  2. jcsd
  3. Sep 29, 2014 #2

    Orodruin

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    You have written down some conservation laws, did you try applying them?
     
  4. Sep 29, 2014 #3
    The thing is I'm not entirely sure how to. I can derive some basic things such as due to conservation of momentum ##m\vec{v}=m\vec{u}+M\vec{U}## and due to the system be isolated ##\frac{1}{2}mv^2=\frac{1}{2}(mv^2+MU^2)##. After this I'm not sure where to go to get the maximum kinetic energy trasnferred.
     
  5. Sep 29, 2014 #4

    BvU

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    You mean ##
    \frac{1}{2}mv^2=\frac{1}{2}(m{\bf u}^2+MU^2)
    ##, right ?

    So there are three equations, but four unknowns. That leaves one degree of freedom.
    What is the kinetic energy you 'want' to maximize? Write it as a function of this one free variable left over.
    Then try to find a maximum for that function.
     
  6. Sep 29, 2014 #5
    Well since the object in question is the first ball, we have to find the a function of ##v## right? We can't use ##u## because that is the velocity after impact.
     
  7. Sep 29, 2014 #6

    BvU

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    No such freedom exists: ##\vec v = (v_x, v_y)## is a given ! ##v_y=0## and ##v_x## will appear in the answer, but as a parameter, not as a variable.
     
  8. Sep 29, 2014 #7
    Could you explain that in more detail?
     
  9. Sep 30, 2014 #8

    ehild

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    The kinetic energy before impact is 1/2 mv2. After impact, it is 1/2 mu2+1/2 M U2, u is the speed of the ball with mass m and U is the speed of the billiard ball of mass M.

    Write up the conservation laws also for the momentum components. Choose a coordinate system with x axis parallel with the initial velocity v of the the small ball and express the components with the angle they enclose with the x axis.

    ehild
     
  10. Sep 30, 2014 #9
    Okay, so when exactly can I find the maximum kinetic energy? I'm thinking if it's when the velocity of m is zero.
     
  11. Oct 1, 2014 #10

    ehild

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    The masses are given, and so is the velocity of the small ball. But the collision is not necessarily central. The direction of the velocities are the free parameters, but the conservation laws make them related, so only one angle is free.

    Find the kinetic energy of the billiard ball after the collision, as function of its angle it encloses with the x axis. You can figure out at what angle is the KE maximum.

    ehild
     
  12. Oct 1, 2014 #11
    [QUOTE="ehild, post: 4868343, member: 481" You can figure out at what angle is the KE maximum.

    ehild[/QUOTE]How exactly would I do that?
     
  13. Oct 1, 2014 #12

    BvU

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    Play with coins n a smooth table to get ideas.
    Draw a picture, then follow ehild's advice. Exactly.
     
  14. Oct 1, 2014 #13
    Okay, but how exactly would one determine the angle where there is maximum kinetic energy transfer. That part doesn't make sense to me. How would a drawing of vectors show that?
     
  15. Oct 1, 2014 #14

    ehild

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    The kinetic energy will depend on some angle. To have maximum, its derivative with respect to the angle should be zero.
    Drawing the velocity vectors help you to write up equations for the momentum components.

    ehild
     
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