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Find the coefficient of restitution

  1. Sep 27, 2012 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    Two equal balls are in contact on a table. A third equal ball strikes them simultaneously and remains at rest after the impact. Show that the coefficient of restitution is 2/3.
    I have attached an image for clarity of problem. Please open it.

    2. Relevant equations
    Conservation of Momentum

    3. The attempt at a solution
    Let the mass of balls be m and coefficient of restitution be e.

    [itex]\frac{v_{2}}{vcosθ_{2}} = e[/itex]

    Also,
    [itex]\frac{v_{1}}{vcosθ_{1}} = e[/itex]

    Applying Conservation of Momentum along X-axis

    [itex]v=v_{1}cosθ_{1}+v_{2}cosθ_{2}[/itex]
    Along Y- axis
    [itex]v_{1}sinθ_{1}=v_{2}sinθ_{2}[/itex]

    Substituting the value of v1 and v2 in the above equation
    [itex]evcosθ_{1}sinθ_{1}=evcosθ_{2}sinθ_{2}[/itex]
    [itex]cosθ_{1}sinθ_{1}=cosθ_{2}sinθ_{2}[/itex]

    Multiplying 2 on both sides
    [itex]sin2θ_{1}=sin2θ_{2}[/itex]
    Rearranging and simplifying
    [itex]2cos(θ_{1}+θ_{2})sin(θ_{1}-θ_{2})=0[/itex]
    [itex]θ_{1}+θ_{2}=\frac{∏}{2}[/itex]
    [itex]θ_{1}-θ_{2}=0[/itex]

    [itex]θ_{1}=θ_{2} and θ_{1} = \frac{∏}{4}[/itex]

    Now substituting the value of θ1 in equation of momentum along X-axis

    [itex]1=e(cos^{2}θ_{1}+cos^{2}θ_{2})[/itex]
    [itex]e=\frac{1}{cos^{2}θ_{1}+cos^{2}θ_{2}}[/itex]
    [itex]e=\frac{1}{2cos^{2}θ_{1}}[/itex]
    [itex]e=1[/itex]

    What's wrong here??? :confused:
     

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  2. jcsd
  3. Sep 27, 2012 #2
    I will say first to reconsider your expressions for the coefficients of restitution. Also check the momentum conservation. Furthermore it may help to consider energy conservation as well. As this gives a further restraint on the values in the problem.
     
  4. Sep 28, 2012 #3

    utkarshakash

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    energy conservation principle can't be applied here because the collision is not elastic. I also suspect something wrong in my expressions for coefficient of restitution. But I can't figure out what is it. Also I don't think that momentum conservation equations are wrong.
     
  5. Sep 28, 2012 #4
    Well if it is indeed an inelastic collision than you are correct. It was not specified in your problem statement. You're off by a minus sign in one of your momentum conservation equations. The definition of the coefficient of restitution is fractional value of the ratio of speeds before and after a collision. You should be able to find the error by just considering this definition and looking at your expressions for, e.
     
  6. Sep 29, 2012 #5
    If the 3 balls are identical and the collisions occur simultaneously. You can immediately deduce that.
    The lines of impacts of both collisions pass through the centres and therefore,
    [tex]\theta_1=\theta_2=\frac{\pi}{6}\\
    and\\
    v_1=v_2[/tex]
    Then your answer is therefore [itex]e=\frac{2}{3}[/itex]
     

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    Last edited: Sep 29, 2012
  7. Sep 29, 2012 #6

    utkarshakash

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    Gold Member

    Hey, why didn't I think it earlier!!!!! Thank You.
     
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