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## Homework Statement

A small object with an initial velocity collides with and sticks to a long thing rod that is free to translate as well as rotate. Motion here is on a horizontal frictionless surface.

Find where the small object would collide with the rod to maximize the loss of energy in the collision and this loss of energy? Explain.

## Homework Equations

## The Attempt at a Solution

So since the rod which the bullet collides into is free to translate and rotate, thus, conservation of linear and angular momentum is conserved since the surface is frictionless.

I'm a little unsure of what equations should I use... Right now i'm using conservation of angular momentum, and the definition of the coefficient of restitution, setting it equal to zero...

So if d is the length of the thin rod, and x is the distance at which the bullet collides with the rod, I can find the angular velocity

[tex] l_0 + L_0 = L_f [/tex]

[tex] xmv + 0 = I_0w[/tex]

Therefore

[tex] w = \frac{xmv}{I_0}\right) [/tex]

Then, I find the velocity using the COR

[tex] 0 = \frac{xw - v_f}{v_0}\right) [/tex]

Therefore

[tex] v_f = xw = x \frac{xmv}{I_0}\right) [/tex]

Finally I set up the loss of K.E. function...

[tex] f(x) = \frac{mv_0^2}{2}\right) - ( \frac{(m+M)v_f^2}{2}\right) + \frac{I_0w^2}{2}\right) ) [/tex]

Substituting the values of v_f, and w into the function, i can differentiate to determine the maximum value of x in order to maximize a loss of K.E. in the system...

My question is am I setting this up right? after differentiating, and setting my derivative equal to zero, I keep getting a negative root... Should I be using different equations?

Any help would be greatly appreciated. :D