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Collison of Billiard Balls

  1. May 31, 2004 #1

    I was wondering if someone can help me with this problem:

    Two biilard balls of equal mass undergo a perfectly elastic head-on collison. If the speed of one ball was initially 2.00 m/s and of the other 3.00 m/s in the opposite direction, what will be their speeds after the collision???????
  2. jcsd
  3. May 31, 2004 #2
    See this:
    In short - the speed of the first ball would be 3 m/s and the speed of the second ball would be 2 m/s.

    Conservation of momentum:
    [tex]m_1v_1 + m_2v_2 = m_1u_1 + m_2u_2[/tex]

    [tex]m_1 = m_2[/tex]

    [tex]v_1 + v_2 = u_1 + u_2[/tex]

    [tex](v_1 + v_2)^2 = (u_1 + u_2)^2[/tex]

    [tex]v_1^2 + 2v_1v_2 + v_2^2 = u_1^2 + 2u_1u_2 + u_2^2[/tex]

    Conservation of energy:
    [tex]\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2[/tex]

    [tex]m_1 = m_2[/tex]

    [tex]v_1^2 + v_2^2 = u_1^2 + u_2^2[/tex]

    So we have:
    [tex]v_1^2 + 2v_1v_2 + v_2^2 = u_1^2 + 2u_1u_2 + u_2^2[/tex]

    [tex]v_1^2 + v_2^2 = u_1^2 + u_2^2[/tex]

    Substract them:
    [tex]2v_1v_2 = 2u_1u_2[/tex]

    [tex]u_1 = \frac{v_1v_2}{u_2} = v_1 + v_2 - u_2[/tex]

    [tex]u_1 = v_1v_2 = v_1u_2 + v_2u_2 - u_2^2[/tex]

    [tex]u_2^2 - (v_1 + v_2)u_2 + v_1v_2 = 0[/tex]

    A quick inspection would reveal that the solutions to that are v1 and v2. v2 is the speed before collision, so we are left with v1. Hence u1 = v2 and u2 = v1. There are probably much easier ways to come to this but it's late. :zzz:
    Last edited: May 31, 2004
  4. May 31, 2004 #3
    I am still confused...I use these equations but it doesnt give me the right answer....
  5. May 31, 2004 #4


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    Science Advisor

    They will interchange speeds. Ball 1 was at 2m/s, after collision is -3m/s. Ball 2 was at -3m/s, after collision is 2m/s.

    Edit: I notice Chen already gave this answer. Here's another derivation:
    [tex]v_1+v_2=u_1+u_2\mbox{ and }v_1^2+v_2^2=u_1^2+u_2^2[/tex]
    Move them around a bit:
    [tex]v_1-u_1=u_2-v_2\mbox{ and }v_1^2-u_1^2=u_2^2-v_2^2[/tex]
    Divide these last two equations and get:
    Subtract this from the first equation and also add it to the first equation, get:
    [tex]v_2=u_1\mbox{ and }v_1=u_2[/tex]
    Last edited: May 31, 2004
  6. Jun 1, 2004 #5
    I knew there was an easier way to get to that. :tongue:
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