Rotational Motion proof Can we mathematically prove that an object lying on the ground, if imparted an impulse (not on the COM) will rotate about its centre of mass and no other axis?
 A: During the collision, which we assume to happen in a short time $\delta t$, the object will receive momentum $\vec{p}$:
$$
\vec{p} = \vec{F} \delta t \quad F \sim \frac{1}{\delta}
$$
And angular momentum $\vec{L}$:
$$
\vec{L} = \vec{r} \times \vec{F} \delta
$$
where $\vec{r}$ is the radius vector from the center of mass to the point, where the impulse was imparted.
After the collision no external force acts on the object, therefore its center of mass has to move with a uniform velocity. If it was rotating around some axis, its motion  would not be uniform.
A: Suppose the object is a non balanced steel roll, the lower half is a little heavier than the upper half.
It is hit by a shot perpendicular to its axis, but well above the center.
By conservation of linear momentum it moves forward, but as the impact is off center, there is a torque, and it also rotates.
But in order to rotate, the COM must turn around the axis of symmetry.
Depending on the force and location of the impact, and the magnitude of the imbalance, it can follow an oscillatory movement, or a rotating movement. But it doesn't rotate around the COM in any case from the perspective of an inertial observer.
A: you have two objects with mass $~m$ and $M$ that collide.
first I write the equations of motions during the collision
$$m\,\frac{dv_1}{dt}=F_c\tag 1$$
$$M\,\frac{dv_2}{dt}=-F_c\tag 2$$
where $F_c$ is the constraint force between $m$ and $M$ .
multiply Eq. (1) (2) with $dt$  and integration you obtain
$$m\,\int_{v_{1i}}^{v_{1f}} dv_1=\int_{ti}^{tf} F_c\,dt=dp$$
$$M\,\int_{v_{2i}}^{v_{2f}} dv_2=-\int_{ti}^{tf} F_c\,dt=dp$$
or
$$m\left(v_{1f}-v_{1i}\right)=+dp\tag 3$$
$$M\left(v_{2f}-v_{2i}\right)=-dp\tag 4$$
and during the collision
$$v_{2f}=v_{1f}\tag 5$$
solving Eq. $~(3)~,(4)~,(5)~$ for $~v_{1f}~,v_{2f}~,dp$ you obtain
$$v_{1f}=\frac {m\,v_{1i}+M\,v_{2i}}{m+M}\tag 6$$
$$v_{2f}=v_{1f}$$
Eq. $~(6)~$  is  the  velocity of the center of mass , thus the objects velocities are the CoM velocity
