A thin spherical shell of radius R , resting on a rough horizontal...

Let the impulse applied by the cue be $J$ at a height $h$ above the centre of the thin spherical shell of mass $M$ and radius $R$.

The linear velocity $v$ acquired by the centre of mass is given by the linear impulse:

$J = Mv$

The angular velocity $\omega$ acquired by the shell is given by the angular impulse about the centre of mass:

$Jh = I\omega$

For a thin spherical shell, the moment of inertia about its diameter is $I = \dfrac{2}{3}MR^2$.

For the shell to not slip on the surface immediately after being struck, the velocity of the point of contact with the ground must be zero, which requires pure rolling:

$v = \omega R$

Substituting $v = \dfrac{J}{M}$ and $\omega = \dfrac{Jh}{I}$ into the pure rolling condition:

$\dfrac{J}{M} = \left(\dfrac{Jh}{\dfrac{2}{3}MR^2}\right)R$

$\dfrac{J}{M} = \dfrac{3Jh}{2MR}$

$1 = \dfrac{3h}{2R}$

$h = \dfrac{2R}{3}$

Thus, the shell should be struck at a height $\dfrac{2R}{3}$ above the centre.