Two solid spheres of radius R and mass m are initially separated by a distance d such that the two centres of mass are separated by d+3R as shown below. The spheres are initially at rest, but the gravitational force  accelerates them towards each other until they collide. a) Calculate the initial mechanical energy of the system. b) Calculate the gravitational potential energy of the system at the moment they collide. c) Derive an expression for v. State carefully what physical law you are applying. d) Show that, in the limit d >> R, the velocity v is independent of d.

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Two solid spheres of radius R and mass m are initially separated by a distance d such that the two centres of mass are separated by d+3R as shown below. The spheres are initially at rest, but the gravitational force  accelerates them towards each other until they collide.
a) Calculate the initial mechanical energy of the system.

b) Calculate the gravitational potential energy of the system at the moment they collide.

c) Derive an expression for v. State carefully what physical law you are applying.

d) Show that, in the limit d >> R, the velocity v is independent of d.

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