A uniform solid sphere of radius $$R$$ produces a gravitational acceleration of $$a_0$$ on its surface. The distance of the point from the centre of the sphere where the gravitational acceleration becomes $$\frac{a_0}{4}$$ is

A projectile is thrown straight upward from the earth's surface with an initial speed $$v=\alpha v_e$$ where $$\alpha$$ is a constant and $$v_e$$ is the escape speed. The projectile travels upto a height 800 km from earth's surface, before it comes to rest. The value of the constant $$\alpha$$ is (radius of the earth $$=6400 \mathrm{~km}$$)

If the Earth stops rotating in its orbit about the sun, there will be variation in the weight of our bodies at

At what depth below surface of the Earth, the acceleration due to gravity will be half of its value that at $$1600 \mathrm{~km}$$ above the surface of the Earth?