In this problem, you will estimate the cross section for an earth-asteroid collision. In all that follows, assume that the earth is fixed in space and that the radius of the asteroid is much less than the radius R of the earth. The mass of the earth is Me, and the mass of the asteroid is m. Use G for the universal gravitational constant. (Figure 1)

Part A
Far away from the earth, the asteroid is moving with speed v and has impact parameter b, as shown in the figure. In this large-separation limit, the distance from the asteroid to the earth is taken to be infinite. Find the total initial energy E_initial of the asteroid.
Express your answer in terms of m, M_e, b, v, and G.

Part B
For large earth-asteroid separation, what is the magnitude of the asteroid's total angular momentum L_initial about the center of the earth?
Express your answer in terms of m, M_e, b, v, and G.

Part C
The maximum impact parameter for which collision is guaranteed, b=b_max , is obtained by setting the minimum earth-asteroid separation equal to the radius R of the earth. This is the configuration shown in the figure. In this case, it is clear that the velocity of the asteroid right before it hits the earth is tangent to the surface and therefore perpendicular to the position vector that points from the center of the earth to the asteroid.
When b=b_max , what is the total energy E_surface of the asteroid the instant before it crashes into the earth? Assume that the speed of the asteroid at closest approach is v_f .
Express your answer in terms of m , M_e , R , v_f , and G .
Hint: What is the gravitational potential energy U_surface of the asteroid when it reaches the surface of the earth?

Part D
Again, suppose that b=b_max. What is the angular momentum L_surface of the asteroid the moment before it crashes into the earth's surface?
Express your answer in terms of m, M_e, R, v_f, and G.

Part E
Use conservation of energy and angular momentum to find an expression for (b_max)^2.
Express your answer as a function in terms of v, R, m, G and Me.
Hint: Use conservation of angular momentum to find v_f .
Express your answer in terms of v , b_max , and R .

Part F
The collision cross section S represents the effective target area "seen" by the asteroid and is found by multiplying (b_max)^2 by π. If the asteroid comes into this area, it is guaranteed to collide with the earth.
A simple representation of the cross section is obtained when we write v in terms of v_e, the escape speed from the surface of the earth. First, find an expression for v_e, and let v=C⋅v_e, where C is a constant of proportionality. Then combine this with your result for (b_max)^2 to write a simple-looking expression for S in terms of R and C.
Express the collision cross section in terms of R and C.
Hint: What is the escape speed v_e at the surface of the earth?
Express your answer in terms of M_e , G , and R .

Part G
The point of origin of a typical asteroid might lie at a radius of about 3 AU (astronomical units; 1 AU=1.5×10^11 m) from the sun, the approximate location of the asteroid belt. Calculate the effective target cross section S of the earth as seen by the asteroid. Assume the asteroid's orbit is circular.
Give your answer as a multiple of the area of the disk of the earth, πR^2.

Hint 1: Here you can find several constants that you may need in your calculations.
Gravitational constant: G=6.67×10^11 N⋅m^2/kg^2
Mass of the Earth: M_e=5.98×10^24 kg
Radius of the Earth: R=6.37×10^6 m
Mass of the Sun: M_s=1.99×10^30 kg

Hint 2: Find the orbital speed v_orbital of the asteroid at a distance of 3 AU from the sun.
Convert your answer to kilometers per second, and round your answer to the nearest tenth place (for instance, 47.28→47.3 ).
Hint for hint 2: What is the magnitude of the centripetal acceleration v^2/r of a particle in a circular orbit of radius r in a gravitational field due to a mass M ?
Express your answer in terms of M , r , and G .