Serway & Jewett — Universal Gravitation: Problem 34

Two spheres having masses \(M\) and \(2M\) and radii \(R\) and \(3R\), respectively, are simultaneously released from rest when the distance between their centers is \(12R\). Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What two isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for \(\vec{v}_1\), the velocity of the sphere of mass \(M\) at any time after release in terms of \(\vec{v}_2\), the velocity of \(2M\). (c) Write an equation from the other model and solve it for speed \(v_1\) in terms of speed \(v_2\) when the spheres collide. (d) Combine the two equations to find the two speeds \(v_1\) and \(v_2\) when the spheres collide.