 Solution

The symbols in the question are diameters: write drum radius $a=R/2$, axle radius $b=r/2$. For a thin hollow drum, $I=M{a}^2$. A rightward horizontal impulse $J$ applied at height $h$ above the centre gives $v=J/M$ and clockwise angular velocity $\omega =Jh/I$. Rolling on the axle requires $v=b\omega$, soThe frictionless rails do not create the required rotation; the initial impulse must give exactly this ratio of linear to angular motion. For the usual case $r<R$, the required height is greater than the drum radius. It cannot be reached by striking the bare cylindrical surface horizontally; an extension or equivalent applied couple would be needed. This geometric limitation should accompany the formal result.