Solving Complex Trajectory Puzzle: Find m's Path Equation

[brupenney]

Can anyone solve this puzzle for me - a mass m in space with a constant velocity C heads toward a circular mass M such that if not disturbed it would pass by M at a distance of 2 of M's radiuses. However, the mass m experiences a second velocity Ve towards the center of M; the magnitude of this V is given by k(d^-1/2) where d is the radial distance to M's center. m starts its journey at infinity and ends up overright the center of M. What is the equation of m's path?

I'm not sure whether or not this is clear. A diagram would be needed ideally.

 

[haruspex]

Can you derive a differential equation or two? (I did. Looked a bit better in polar than in Cartesian, but still rather nasty.) Btw, is the undeviated miss by two radii from the centre or from the surface? And what is "overright"?

 

[brupenney]

From the center. I cannot get anywhere with this problem

 

[brupenney]

By overright I mean at an angle from the deviated path or undeviated path to the center of M is such that their separation is the smallest it can be.

 

[haruspex]

Taking the centre of M as the origin in polar co-ordinates, and θ=0 being the undeviated direction:
- what is the velocity when at (r, θ)
- what does that give you for $\dot{r}$ and $\dot{\theta}$

 

[brupenney]

If I understand your question, the velocity at (r,theta) is the vector sum of C and Ve at an angle theta.

I know some calculus, integral and differential, but I just can't figure how to approach this problem, and I may not have enough knowledge to solve it even then.

 

[haruspex]

If I understand your question, the velocity at (r,theta) is the vector sum of C and Ve at an angle theta.

OK, so what is the velocity in the radial direction? In the tangential direction? How do these relate to r-dot and theta-dot?