Question

Two uniform solid spheres of equal radii $R$, but mass $M$ and $4$ $M$ have a centre to centre separation $6$ $R$, as shown in Figure. The two spheres are held fixed. A projectile of mass $m$ is projected from the surface of the sphere of mass $M$ 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.

Answer 1

Two uniform solid spheres

Radius R

Center to center separation = 6R

Mass of Sphere 1 = M

Mass of Sphere 2 = 4M

Kinetic energy = 1/2 m v^2

Where,

v - the velocity of the object and

m - mass.

Hence,

To ( GMM / R ),

Mass + gravitational potential energy

Ei = Kinetic energy - GMM / R - G 4mm / R

v = 0,

Kinetic energy = 0

n = -GMM ​​/ 2R - G 4mm / 4R

ei = n

( 1/2 ) m v - GMM / R - G4mm / R = - GMM ​​/ 2 R - G4mm / 4R

v = ( 3 GM / 5R ) 1/2

Hence, v = ( 3 GM / 5R ) 1/2 is the expression for minimum speed v of the projectile so that it reaches the surface of second sphere.