# Complex trajectory

1. Oct 25, 2012

### 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.

2. Oct 26, 2012

### 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"?

3. Oct 27, 2012

### brupenney

From the center. I cannot get anywhere with this problem

4. Oct 27, 2012

### 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.

5. Oct 27, 2012

### 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 $\dot{\theta}$

6. Oct 27, 2012

### 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 cant figure how to approach this problem, and I may not have enough knowledge to solve it even then.

7. Oct 28, 2012

### haruspex

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