A ball of mass $m$ travelling with a velocity $v_0$ collides elastically, and perpendicularly with a rod at a distance $d$ from the center of the rod. The rod has a mass $M$ and lies on a frictionless table.
Show that the relative velocity between the ball and the hitting point doesn't change because of the collision.
The rod length is not stated in the problem.
I tried to use the 2 following equations: $$m\vec{v_0}=m\vec{u_1}+M\vec{u_2} \\ md\vec{v_0}=md\vec{u_1}+\frac{1}{12}ML^2\cdot \vec{\omega}$$ where the origin is located at the rod's center, I added a new variable $L$, the rod's length, and $\vec{u_1}$ and $\vec{u_2}$ are the ball's and C.M of the rod velocities after the collision, respectivly. I am not sure whether I can use another eq., related to elastic collisions: $$\vec{v_0}+\vec{u_1}=\vec{u_2}$$ Using the whole 3, I wouldn't manage to show that $$\vec{u_0}=\vec{\omega }d$$ (Which is the condition I need to show).
Is there anything incorrect about my reasoning?
