Motion of object under central force

[myron]

Hi everybody

I have been trying to develop an intuitive general understanding of how the trajectory of an object , that is moving at constant velocity, is changed by the sudden appearance of a central force but I am not sure my reasoning is correct.

(1) Consider an object "A" not subject to any force and, therefore, moving at constant velocity "V".

(2) Now, suppose that, "by magic", a central force appears at a distance R from "A" constantly pointing to a point "O" in space.

As far as I understand the path that A will take depends from its velocity "V" at the time the central force appeared:

(a) If V=0, then A will start to oscillate around O

(b) if V= SQR(a x R) (where: SQR = square root, "a" = the acceleration, constant in magnitude, to which A is subject) then A will start to move along a circular path

(c) if V < SQR(a x R), then A will start moving along an elliptical path (the smaller V the more "sqeezed" the ellipse -- the case V=0 could be considered as the degenarate case of this)

(d) if V > SQR(a x R), then the trajectory of A will be curved by the central force but the path will not be (necessarily ?) a "closed" one like a circle or an ellipse

Is the reasoning above correct? Points (a), (b), (c) make sense to me, so I believe they are correct, but I do have strong doubts about point d ...

Thanks a lot for your help!

 

[haruspex]

You haven't specified how the magnitude of the force varies, e.g. a function of R.
For an elliptical path to result, you need a force like gravity, G/R^2. In that case it will be elliptical provided V < escape velocity, sqrt(2aR).