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# Two uniform solid spheres of equal radii R and mass M and 4M have a centre to centre separation of 6R. 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 second sphere. Obtain an expression for the speed when it strikes the heavier sphere of mass 4M.

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Solution

## Two uniform solid sphereRadius RCentre to centre separation =6RMass of sphere 1=MMass of sphere 2=4MA projectile of mass m is projected from the sphere of mass M is directly towards centre of the second sphere.By formula Kinetic energy =12mv2Wherev= velocity of the object and m= massHence,To find the gravitational potential energy potential (GMM/R)Mass + gravitational potential energySubstituting the valuesWe getE1=Kineticenergy−GMM/R−G4MMRConsider v=0ThereforeKinetic energy=0Substituitingn=−−GMM2R−G×4MM4RBy the law of conservation of energyR1=nHence,(12)mv−GMMR−G4MMR=−GMM2R−G4MM4RCalculate vv=3GM/5RI×12v=2GM5R×12

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