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$V(z)=\frac{\sigma}{2 \in_0}\left(\sqrt{R^2+z^2}-z\right)$
A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z=z_0$ and $z_0>0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec{F}=-c \hat{k}$ with $c>0$. Let $\beta=\frac{2 c \in_0}{q \sigma}$. Which of the following statement($s$) is(are) correct?
$(A)$ For $\beta=\frac{1}{4}$ and $z_0=\frac{25}{7} R$, the particle reaches the origin.
$(B)$ For $\beta=\frac{1}{4}$ and $z_0=\frac{3}{7} R$, the particle reaches the origin.
$(C)$ For $\beta=\frac{1}{4}$ and $z_0=\frac{R}{\sqrt{3}}$, the particle returns back to $z=z_0$.
$(D)$ For $\beta>1$ and $z_0>0$, the particle always reaches the origin.
