A Rolling Bat Struck by a Ball

[A. Aspart]

A bat lies along the x-axis so that the tip of its handle is located in (0, 0) while the other end is at (0.9 m; 0). The center of mass is at (0.6 m; 0). A ball hits the bat perpendicularly at (x, 0), causing an impulse I.
The bat’s moment of inertia through the center of mass is J = 0.0530 kgm^2; the bat’s mass m = 0.800 kg.
Calculate x so that the tip of the handle stays at (0, 0) by studying the movement of the center of mass.

I assumed that the axis is parallel with z-axis and that the bat starts rolling around the point (0, 0). I calculated J about the tip of the handle. After that, however, I am stuck since I can only form equations with several unknown variables. Also, I don’t really understand the conditions under which the tip of the handle remains at (0, 0). Have I approached the assignment incorrectly or am I missing something (or both)?

All help is much appreciated.

 

[collinsmark]

I think the problem is asking for the distance x, such that the bat handle's tip's instantaneous velocity is zero, immediately after the impulse.

I can guarantee you that the tip of the handle will not stay stay at (0, 0) indefinitely, without other forces involved holding it in place. But its initial, instantaneous velocity can be zero so long as the x is just right. I think that's what you are being asked to calculate.

[Hint: you need to figure out the situation that the rotation movement of the tip, [STRIKE]ωx[/STRIKE] [Edit: I meant ωx_c, where x_c is the distance from the center of mass to the handle's tip, not to be confused with x, the distance from the tip to the location of the ball striking] is equal and opposite v, the velocity of the center of mass.]