Meteor Entering Orbit of a Sun (cons. energy & momentum) [very hard algebra]

DD31

1. The problem statement, all variables and given/known data
Alright, so I've given this guy a few cracks, and I think I'm close to it, but apparently I'm not right, so here's how it goes. Simple in concept.

A meteor moves toward the solar system with speed v0 in a direction such that it would miss the sun by a distance d if it were not attracted by the sun's gravitational force. Denote the mass of the sun by M. Find the distance b of the meteor from the sun at the point of closest approach in terms of v0, d, and M (and G, the gravitational constant).





2. Relevant equations
They effectively give me a few formulas in the helper questions provided:

The momentum and energy are conserved
Initial energy = 1/2mv0^2
Final energy = 1/2mV^2 - GMm/b

Initial Momentum = mv0d
Final Momentum = mVb

Where V=the final velocity of the object at point b

3. The attempt at a solution

I feel like the error here is most likely in my algebra, as I'm having a ton of troble easily isolating b. But here are the ways I've tried so far:

I always start with the equations:

1/2mv0^2 = 1/2mV^2 - GMm/b

and

mv0d = mVb.

I then cancel out all the m's and multiply out the 2 in the top one to get:

v0^2 = V^2 - 2GM/b

and

v0d = Vb

From this point, I'll either solve for b in the top or solve for V in the bottom. If I solve for b in the top, I get:

V^2 - v0^2 = 2GM/b, b = 2GM/(V^2 - v0^2)

Then I would solve for V in the bottom one:

V = v0d/b

Subbing that in gets

b = 2GM/((v0d/b)^2 - v0^2)

And this is where I think I go wrong...trying to get b out of this. I've tried rearranging the equation to bring all the b's to one side, I've tried resubstituting stuff...most recently I tried quad formula:

I factored out the v0's on the bottom:

2GM/((v0^2)(d^2/b^2-1)) = b

b(d^2/b^2 - 1) = 2GM/v0^2

d^2/b - b = 2GM/v0^2

d^2 - b^2 = 2GMb/v0^2

b^2 + 2GMb/(v0^2) - d^2 = 0

then, using 1 as (a), 2GM/v0^2 as (b) and -d^2 as (c), I did quad form and got:

(-2GM/(v0^2)+sqrt((2GM/v0^2)^2-4d^2))/2

or





But that was not correct


I'm officially stumped by this point...I've taken 4 cracks at it, to no avail. Am I setting up the formulas incorrectly, or am I just screwing up on the algebra over and over?

Redbelly98

You are really close.

What is "-4ac", when c is -d²? (And a=1 of course)

DD31

haha...wow. I see it...it just should be + 4d^2. Just put that in and it worked. Thanks a ton...Almost sure I wouldn't have picked that out without doing the whole thing over again.

Thanks.