3. A small moon of mass m is in a circular orbit of radius R around a planet of mass M >> m. The kinetic energy of the moon is (1/2)mv and the potential energy is -k/R, where k = GMm. (a) Find the velocity, v;, of the orbiting moon in terms of m, R, and k = GMm. We can describe any orbit by the formula r = 1 + E cos 0 Find a and e for this orbit. (b) Now an asteroid with the same mass as the moon and half the velocity has a head on collision with the moon (to be explicit, if the velocity of the moon is v at the point of collision, the velocity of the asteroid is -v/2). The collision is instantancous, and after the collision, the asteroid and the moon combine so that right after the collision we have a mass of 2 at the place the moon was right before the collision. Use conservation of momentum to find the velocity of the combined masses after the collision. (c) Calculate the new orbital parameters for the combined masses. In particu- lar, find of, the value of a after the collision, and ey, the value of c after the collision. For a your answer should be of the form of a number times R, and for ef you should be able to find an actual number