# Ball hitting plank

1. Nov 22, 2009

### Knissp

1. The problem statement, all variables and given/known data
A plank of length 2L and mass M lies on a frictionless plane. A ball of mass m and speed Vo strikes its end (the plank is standing vertical and the ball strikes the top from the left). Find the final velocity of the ball, Vf, assuming that mechanical energy is conserved and that Vf is along the original line of motion.

2. Relevant equations
Conservation laws (angular momentum, energy)

3. The attempt at a solution
I am having conceptual difficulty here. I know that initially angular momentum (about the center of the rod) is given by $$\widehat{L_0} = m V_0 L$$. After impact, angular momentum is $$\widehat{L_f} = m V_f L + I \omega$$. Conservation of energy gives $$1/2 m V_0^2 = 1/2 m V_f^2 + 1/2 I \omega^2$$. Given I, I can solve for $$\omega$$ and substitute and do some algebra to find $$V_f$$ (which I have already done but don't want to type; it's not relevant to my question). I used $$I = 1/12 M (2L)^2 = 1/3 M L^2$$, which is the moment of inertia about the center of the rod. I am not sure, however, if it is valid to assume that the rod rotates about its center. I feel as if this is intuitively true but am not sure how to completely convince myself of this, so perhaps I am incorrect. Also, I'm assuming that the rod does not translate. Is this a valid assumption? Does anyone have any input?

EDIT: The hint for this question says if m=M, then Vf = 3/5 Vo. The technique I used does not yield that result, so I must be setting this up incorrectly. This tells me that at least one of my assumptions is flawed. Any ideas on which one?

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Last edited: Nov 22, 2009