Escape velocity is given by \(v_e = R\sqrt{\frac{8}{3}\pi G\rho}\). Thus, \(v_e \propto R\sqrt{\rho}\). The ratio is \(frac{v_e}{v_p} = \frac{R_e}{R_p}\sqrt{\frac{\rho_e}{\rho_p}} = \frac{1}{2}\sqrt{\frac{1}{2}} = \frac{1}{2\sqrt{2}}\).

Assertion (A) is true. Gravitational potential at the center of a uniform solid sphere is finite (e.g., \( -\frac{3}{2} \frac{GM}{R} \)), hence potential energy is non-zero. Reason (R) is true. Due to symmetry, the net gravitational field at the Earth's center is zero. However, zero field does not directly explain non-zero potential energy. Thus, (R) is not the correct explanation of (A).

Assertion (A) is true.

Escape velocity \( v_e = \sqrt{\frac{2GM}{R}} \). Substitute \( M = \rho \frac{4}{3} \pi R^3 \) to get \( v_e = \sqrt{\frac{8G\pi R^2 \rho}{3}} \). If \( R^2 \rho \) is constant (derived from \( A\rho \) being constant), then \( v_e \) is constant. Reason (R) is true, as \( v_e = \sqrt{\frac{2GM}{R}} \) shows \( v_e \propto R^{-1/2} \) for constant \( M \). However, (R) does not explain (A), as the conditions are different.

Assertion (A) is true: escape velocity \(V_e = \sqrt{2GM/r}\) is \(\sqrt{2}\) times orbital velocity \(V_o = \sqrt{GM/r}\) for a circular orbit.

Reason (R) is false because the gravitational field extends infinitely. Escaping means overcoming the gravitational potential, not leaving the field.

Assertion (A) is true; the gravitational potential energy is more negative (deeper well) inside a uniform planet, requiring greater escape velocity.

Reason (R) is true and foundational: gravity being a conservative central force allows the definition and calculation of potential energy and escape velocity.