# Inelastic launch

1. Jun 6, 2010

### benf.stokes

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 $$\lambda$$
, is thrown vertically up with velocity $$v_{0}$$.
Show that the maximum height that M can reach is:

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

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

2. Relevant equations

$$F=\frac{dp}{dt}=\frac{dp}{dx}\cdot v$$

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:
$$M_{total}=M+\lambda\cdot y$$ and thus the momentum at any position is given by:

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

$$F=\frac{dp}{dt}$$ I get an differential equation I can't solve.

Any help would be appreciated.

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