 Solution

Initially $r_1=a$ and $z_2=\sqrt{2}a-l$, so the total gravitational potential energy isIf the given velocity is understood as a full tangential vector $\mathbf{v}$, put $v_z=\mathbf{v}\cdot {\mathbf{e}}_z$. The string length constraint gives ${\dot{z}}_2(0)=-\sqrt{2}{v}_z$, and henceThis formula accounts for both particles' kinetic energies, including that of the hanging particle.

If $v$ denotes only the initial speed, the printed wording does not specify its direction within the tangent plane. To make that dependence explicit, let $\beta$ be the angle from the downward generator of the cone towards the azimuthal unit vector, soThen $v_z=-v\cos \beta /\sqrt{2}$, andFor a launch along a generator, $\beta =0$ or $\pi$, this reduces to $E=\frac{1}{2}(m_1+m_2)v^2-m_{1}ga+m_{2}g(\sqrt{2}a-l)$. For a horizontal azimuthal launch, $\beta =\pi /2$, it instead gives $E=\frac{1}{2}{m}_1{v}^2-m_{1}ga+m_{2}g(\sqrt{2}a-l)$. A scalar speed alone does not determine a unique energy here. Both directions are tangent to the surface; “along the surface” does not distinguish them. The assumed configuration also requires $l>\sqrt{2}a$ so the second particle hangs below the apex.