Energy imparted to Rod due to Constant Impulse

On a frictionless table is a rod of length = L, and of uniform density, mass = M. It is shown below with two points marked on it. One is "c", the physical center of the rod (i.e., happens to be its CM), and the other is p, an arbitrary point on the rod. (Let d == the distance between p and c, and we'll say that d>0, so that any rotation is clockwise.)

A horizontal impulse, I, is applied to the rod at point p. The rod is initially not moving and its KE=0. Assuming no friction, then after the impulse, the rod's total KE = KE_t + KE_r (i.e., the sum of the translational and rotational KE).

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p -----------> I
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c
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If p happens to be c, then KE_r = 0; otherwise KE_r > 0. The further p is away from c, the greater will be the angular velocity around c, and so, the greater is KE_r.

My first belief was that for a constant I, the resultant KE would be a constant regardless of where p was. But now I am not so sure since if p=c, then KE = KE_t, since there is no rotation.

So, to help clear this confusion, I would like to understand better the relationship in formulas between a constant I (impulse) applied to different points, p, and the resultant total energy of the rod.

I was thinking about other related problems when I realized that I wasn't sure about the relationship between constant impulse and change in KE.