A solid sphere of mass ' m ' and radius ' r ' is allowed to roll without slipping from the highest point of an inclined plane of length ' L ' and…

A solid sphere of mass ' $\displaystyle m$ ' and radius ' $\displaystyle r$ ' is allowed to roll without slipping from the highest point of an inclined plane of length ' $\displaystyle L$ ' and makes an angle $\displaystyle 30^{\circ}$ with the horizontal. The speed of the particle at the bottom of the plane is $\displaystyle v_1$. If the angle of inclination is increased to $\displaystyle 45^{\circ}$ while keeping L constant. Then the new speed of the sphere at the bottom of the plane is $\displaystyle v_2$. The ratio $\displaystyle v_1{ }^2: v_2{ }^2$ is