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## Homework Statement

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 M_e, and the mass of the asteroid is m. Use G for the universal gravitational constant.

I've already solved:

E_initial = (m/2)v

^{2})

L_initial = bmv

E_at surface of Earth = ((v_f)

^{2}*m)/2 +(GmM

_{e})/R

L_at surface of Earth = (mv

_{f}R

b

_{max}

^{2}= R

^{2}+(2RGM

_{e})/v

^{2}

Where I get confused is the actual significance of b

_{max}:

The collision cross section S represents the effective target area "seen" by the asteroid and is found by multiplying (bmax)

^{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 ve, the escape speed from the surface of the earth. First, find an expression for v

_{e}, and let v=Cv

_{e}, where C is a constant of proportionality. Then combine this with your result for (bmax)

^{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.

I know that a version of v

_{e}is sqrt(2GM/R), but I have no idea how to go on from here.