= Solution
The symbols in the question are diameters: write drum radius $a=R/2$, axle radius $b=r/2$. For a thin hollow drum, $I=Ma^2$. 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 $v=b\omega$, so
$$
\boxed{h=\frac I{Mb}=\frac{a^2}{b}=\frac{R^2}{2r}.}
$$
The 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.
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