An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth. Determine the height of the satellite above the earth's surface. 

Two uniform solid spheres of equal radii $\mathrm{R}$, but masses $\mathrm{M}$ and $4 \mathrm{M}$ have a centre to centre separation of $6 \mathrm{R}$, as shown in Fig. $12.53$. The two spheres are held fixed. A projectile

of mass $\mathrm{m}$ is projected from the surface of the sphere of mass $\mathrm{M}$ directly towards the centre of the second sphere. Obtain an expression for the minimum speed $v$ of the projectile so that it reaches the surface of the second sphere.