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Kinetic enengy needed to break a board

  1. Jan 29, 2013 #1
    1. The problem statement, all variables and given/known data

    I need to get [itex]E_{b}[/itex] = .5(m*n*[itex]V^{2}_{i}[/itex])/(m+n) [eq 6]

    where [itex]E_{b}[/itex] is the energy required to break the board, m is the mass of the object hitting the boards, n is the mass of the board and [itex]V_{i}[/itex] is the velocity of the object before hitting the board.

    2. Relevant equations

    [itex]K_{i}[/itex]=.5*m*[itex]V^{2}_{i}[/itex] [eq 1)
    where [itex]K_{i}[/itex] is the initial kinetic energy, m is the mass of the object, [itex]V_{i}[/itex] is the inital velocity

    [itex]K_{f}[/itex]= [itex]E_{b}[/itex] + .5*(m+ n)*[itex]V^{2}_{f}[/itex] [Eq2]
    [itex]E_{b}[/itex] is the energy required to break the board, m is the mass of the object hitting the boards, n is the mass of the board and [itex]V_{i}[/itex] is the velocity of the object before hitting the board.

    momentum before = momentum after
    m8[itex]V_{i}[/itex] = (m+n)*[itex]V_{f}[/itex]

    [itex]V_{f}[/itex]=(m*[itex]V_{i}[/itex])/(m+n) [eq3]


    3. The attempt at a solution

    I know make eq 1 and 2 equal (assuming no energy is lost) rearrange the equation to get E on its own

    [itex]E_{b}[/itex] = .5*m*[itex]V^{2}_{i}[/itex] - .5*(m+ n)*[itex]V^{2}_{f}[/itex] [eq4]

    Now i can sub in eq3 into eq 4

    [itex]E_{b}[/itex] = .5*m*[itex]V^{2}_{i}[/itex] - .5*(m+ n)*[(m*[itex]V_{i}[/itex])/(m+n)[itex])^{2}[/itex]

    [itex]E_{b}[/itex] = .5*m*[itex]V^{2}_{i}[/itex] - .5*[(m*[itex]V_{i}[/itex][itex])^{2}[/itex]/(m+n)]

    This is where I am stuck, I cannot get to equation 6.
    Can anyone help me?
     
  2. jcsd
  3. Jan 29, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    ##E_{b} = \frac{1}{2}mV^{2}_{i} - \frac{1}{2}\frac{(mV_{i})^{2}}{(m+n)}## - this is your last equation

    ##E_{b} = \frac{1}{2}mV^{2}_{i}\frac{m+n}{m+n} - \frac{1}{2}\frac{mmV_{i}^{2}}{(m+n)}##
    ##E_{b} = \frac{1}{2}mV^{2}_{i}\frac{m+n-m}{m+n}##
    And the next step is the final result.
    The idea was just to combine both terms.
     
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