Answer Happy

3. Collision Speed (15 pts). Suppose a dwarf planet with the size and mass of Ceres were to enter an elliptical orbit that takes it from the asteroid belt at 3 AU (take this to be aphelion) to the orbit of Mercury at 0.4 AU (take this to be perihelion). (a) [5 pts) How fast would the intruder planet be moving when it reached Mercury's orbit? Assuming both the intruder and Mercury orbit in a prograde direction, and approximating Mercury's orbit as circular, calculate the difference between the intruder's orbital speed and Mercury's orbital speed when they meet. (b) [5 pts] As the two bodies approach each other before collision, let's briefly neglect the presence of the Sun and examine the collision in the center-of-mass frame of the two bodies. Calculate the speed of each body relative to the center of mass, and use these speeds to estimate the total kinetic energy before collision, ignoring any acceleration from the two bodies attracting each other. Which body makes the larger contribution to the total kinetic energy? (c) [5 pts] Use the expressions given in Equations (15.16) and (15.17) of the textbook to determine the relative collision speed including the gravitational energy between the two bodies. Compare your result with the result of part (a), which provides the relative speed at large separation.