(Solved): thank you!!!! In this imaginary problem, an asteroid is initially spotted very far away but travelin ...

In this imaginary problem, an asteroid is initially spotted very far away but traveling in the general direction of Jupiter, in such a way that if it continued to move on a straight line it would miss it by \( r_{\perp}=125000 \mathrm{~km} \) at the point of closest approach (measured from the center of the planet). The gravity of Jupiter bends this trajectory. so the asteroid actually ends up passing much closer, a mere \( 14000 \mathrm{~km} \) above the surface of Jupiter. The radius of Jupiter is \( 71490 \mathrm{~km} \) and its mass is \( 1.9 \times 10^{27} \mathrm{~kg} \). (a) Draw a diagram showing the projected straight-line trajectory as a dashed line, and indicate on it (1) the initial position of the asteroid and (2) what would have been the point of closest approach. Show also the actual (curved) trajectory and the actual point of closest approach (not necessarily to scale!). (b) Write an expression for the initial angular momentum of the asteroid (about the center of Jupiter) and the initial total energy of the system, involving the (unknown) initial velocity, \( v_{i} \), of the asteroid. Assume that at the time the asteroid is initially spotted the effect of Jupiter's gravity is still negligible. (c) Using conservation of angular momentum and of the total energy, find the initial speed of the asteroid, and its speed at the actual point of closest approach. (d) Under these assumptions, is the astepid bound gravitationally to Jupiter (that is to say, on a closed orbit around Jupiter), or not?