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Inelastic launch

  1. Jun 6, 2010 #1
    1. The problem statement, all variables and given/known data
    A mass M attached to an end of a very long chain of mass per unit length [tex]\lambda[/tex]
    , is thrown vertically up with velocity [tex]v_{0}[/tex].
    Show that the maximum height that M can reach is:

    [tex]h=\frac{M}{\lambda}\cdot \left [ \sqrt[3]{1+\frac{3\cdot \lambda\cdot v_{o}^{2}}{2\cdot M\cdot g}}-1 \right ][/tex]

    and that the velocity of M when it returns to the ground is [tex]v=\sqrt{2\cdot g\cdot h}[/tex]

    2. Relevant equations

    [tex]F=\frac{dp}{dt}=\frac{dp}{dx}\cdot v[/tex]

    Conservation of energy cannot be used because inelastic collisions occur in bringing parts of the rope from zero velocity to v
    3. The attempt at a solution

    I start by setting up that the total mass at a position y is:
    [tex]M_{total}=M+\lambda\cdot y[/tex] and thus the momentum at any position is given by:

    [tex]p=(M+\lambda\cdot y)\cdot v[/tex] but I can't figure out an expression for v and using

    [tex]F=\frac{dp}{dt}[/tex] I get an differential equation I can't solve.

    Any help would be appreciated.
     
  2. jcsd
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