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If a bar is floating in space (at rest in the frame) and a point mass strikes the bar perpendicular to the length of the bar and in the exact middle then you can you use the conservation of momentum/kinetic energy to calculate the movement of the bar, its velocity after the impact.

If the point mass strikes the bar perpendicular to the bar at the end of the bar, then the bar will move differently than in the first case. The center mass of the bar will move differently in each of the cases so you can't just know how the center mass of the bar will move by just applying the force to the center of mass.

I think that in the second case the bar will rotate as well as translate so the c.m. can't move in the same way as in the first case becasue then it will have more kinetic energy i.e. the kinetic energy of translation plus the kinetic energy of rotation.

A little fuzzy on this.