According to the principle of conservation of mechanical energy, the total energy of a body moving under an external conservative force (like gravity) remains constant . When an object is shot up a smooth incline, its initial kinetic energy at the bottom is entirely converted into gravitational potential energy ($V = mgh$) at the highest point reached .

1. Energy Relation: For a mass $m$ with projection velocity $v$, the maximum height $h$ reached is given by $\frac{1}{2}mv^2 = mgh$, which simplifies to $h = \frac{v^2}{2g}$.
2. Incline Trigonometry: On an inclined plane, the vertical height $h$ is related to the distance $x$ travelled along the plane by $h = x \sin\theta$.
3. Equating Both Cases: Since the object is shot with the same velocity $v$ in both cases, the vertical height $h$ reached remains the same irrespective of the angle . $x_1 \sin 60^{\circ} = x_2 \sin 30^{\circ}$ $x_1 \left(\frac{\sqrt{3}}{2}\right) = x_2 \left(\frac{1}{2}\right)$ $\frac{x_1}{x_2} = \frac{1}{\sqrt{3}}$

Thus, the ratio $x_1:x_2$ is 1:$\sqrt{3}$.