Motion of a Particle: Solutions & Examples

[Einstenio]

Homework Statement

Show that a point with acceleration given by:
a=c*((dr/dt)×r)/|r|3
where c is a constant, moves on the surface of a cone.

Relevant Equations

v=dr/dt

This is jut an example to illustrate my doubt. I don't know how to obtain the tracjectory given only the acceleration in this format. I realized that if i can show that there is an constat vector 'a' that satisfy a•r=constant, than the motion would be on the surface of a cone. So i tried to make use of some vectorial identity multiplying by cross product on both sides and try to use the 'BAC-CAB' rule, but that didnt lead to anywhere.

Is there any example similar to this case or anywhere i can study to have a better understanding?

 

[haruspex]

if i can show that there is an constat vector 'a' that satisfy a•r=constant, than the motion would be on the surface of a cone.

Seems to me that would be motion in a plane normal to $\vec a$.

 

[haruspex]

$\ddot{\vec r}=c\frac{\dot{\vec r}\times\vec r}{|r|^3}$?
Seems most unlikely that would give a cone. A cone's axis has an orientation in space, whereas that equation appears to have spherical symmetry.