= Solution
Choose the cylinder centre's initial position as the coordinate origin, with downhill motion to the right and $0\le\alpha<\pi/2$. For <rolling without slipping>, the clockwise <angular speed> is $\omega=v/r$. A point initially vertically above the axis therefore has
$$
\boxed{x(t)=vt\cos\alpha+a\sin(vt/r),\qquad
y(t)=-vt\sin\alpha+a\cos(vt/r).}
$$
A different origin adds constant coordinates without changing the following <velocity> test. Setting $\theta=vt/r$ gives $\dot x/v=\cos\alpha+(a/r)\cos\theta$ and $\dot y/v=-\sin\alpha-(a/r)\sin\theta$.
For strictly upward motion, $\dot x=0$ and $\dot y>0$. The first condition fixes $\cos\theta=-(r/a)\cos\alpha$; choose the negative sine branch to maximize $\dot y$. Positivity then requires
$$
\sqrt{(a/r)^2-\cos^2\alpha}>\sin\alpha,
\qquad\text{equivalently}\qquad\boxed{a>r.}
$$
At $a=r$ the relevant point is momentarily stationary, not moving upward. Thus a material point inside or on an ordinary cylinder cannot move exactly vertically upward in this motion. The mathematical extension $a>r$ describes the looping <trochoid> of a point beyond the rolling radius, illustrated in the figure.
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