I Classical Mechanics - Motion of a particle

[Einstenio]

TL;DR Summary

How to find the trajectory given the acceleration as a cross product of velocity and acceleration

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.

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?

 

[anuttarasammyak]

Do you mean $\mathbf{r}$ is confined on the surface of a cone ? If so as an obvious case bodies moving on generatrix of a cone keep constant speed. Bodies moving around circumference of bottom of a cone gets acceleration leaving the cone surface. There should be a confinement to keep them on the surface.