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I've got this question that I just can't seem to answer. The question is as follows:

A comet approaches the solar system with velocity v and would, if the sun would not attract the comet, pass the sun at a distance d. What is in reality the shortest distance between comet and sun? (Hint: use conservation of angular momentum and energy).

I've tried a few things, but apparently they are all wrong. I thought about getting a formula for the distance to the sun, and then finding where the derivative is 0. Is that the right approach? I just can't get to a correct formula. I've also tried equating the total energy and momentum with gravity to the total energy and momentum without gravity, but that becomes a complete mess. Any help or pointers would be greatly appreciated!

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  • $\begingroup$ The Sun will ALWAYS attract the comet, even if the comet doesn't end up hitting the Sun. Your question is really confused: what is it you'd like to know or understand? $\endgroup$
    – Gert
    Commented Nov 18, 2015 at 3:38
  • $\begingroup$ I think you should consider changing the title of your question. Its not so much about having/not having gravity. The parameter, $d$, is the impact parameter, which defines the initial angular momentum to be $L=mv_0d$. You can also consider your system's total energy at this point to be $E_{tot} = 1/2mv_0^2$. $\endgroup$
    – tmwilson26
    Commented Nov 18, 2015 at 4:30
  • $\begingroup$ Hi and welcome to the Physics SE! Please note that this is not a homework help site. Please see this Meta post on asking homework questions and this Meta post for "check my work" problems. $\endgroup$
    – John Rennie
    Commented Nov 18, 2015 at 7:24
  • $\begingroup$ Similar to physics.stackexchange.com/questions/127646/… $\endgroup$
    – BowlOfRed
    Commented Nov 18, 2015 at 7:51

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Its not that tough. You can work it out by using just two equations. But the one thing you should keep in mind is that when the comet is at the minimum distance from the sun, its velocity must be perpendicular to the radial vector (sun to comet). So the minimum distance is itself the minimum perpendicular distance used in the angular momentum formula at that point. This is so, because of the fact that its locus must be derivable at every point. You will arive at a quadratic equation, solve it to get minimum distance. (you should also know the average diametre of the solar system, to do this)

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