A disk of radius \( R \) with uniform positive charge density \( \sigma \) is placed on the \( xy \) plane with its center at the origin. The Coulomb potential along the \( z \)-axis is \( V(z)=\dfrac{\sigma}{2\epsilon_{0}}\left(\sqrt{R^{2}+z^{2}}-z\right) \). 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 \( \beta=\dfrac{2c\,\in_{0}}{q\sigma} \). Which of the following statement(s) is(are) correct?
