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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.

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Solution

## 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.

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