In article ,
writes:
How can I show that the gravitational impact parameter for e.g. an
asteroid to hit earth is

S = sqrt(R^2 + 2GMR/V^2)

where M and R are the Earth's mass and radius, and the asteroid is
moving very far away with speed V?

I don't think you have to solve any differential equations. Just
note that both energy and angular momentum are conserved. Write down
the respective quantities at both "infinity" and at closest approach
to earth, equate them, and solve for the impact parameter at infinity
when the closest approach distance is one earth radius.

You could also use the virial theorem instead of conservation of
angular momentum, or at least I think you can.

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