# What can you say about the motion of an object with velocity vector perpendicular to position vector

What can you say about the motion of an object with velocity vector perpendicular to position vector? Can you say anything about it at all?
I know that velocity is always perpendicular to the position vector for circular motion and at the endpoints of elliptical motion. Is there a general statement that can be made about the object's motion when the velocity is perpendicular to position?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

regulerenes4w

If you are constrained to move in a plane, then yes, circular motion is the only possibility. To see this, let (x(t),y(t)) be the coordinates at time t on the path. The condition is

Integrate with respect to t:
$\int x\frac{dx}{dt}dt+\int y\frac{dy}{dt}dt=\int 0,dt$

$\frac{{x}^{2}}{2}+\frac{{y}^{2}}{2}=C$
which is the equation of a circle.
So your path has to move around the circle. But that doesn't mean the path itself is a circle. It can speed up. It can slow down. It can stop and reverse direction, all without violating the perpendicularity condition (when it is stopped, the velocity is 0, whose inner product with the position vector is still 0 - this stretches the definition of "perpendicular" a bit, but generally it's allowed, as the physical objects being modelled occasionally do stop).
In three dimensions, a similar calculation shows you that the motion is constrained to some sphere. But you can move in all sorts of ways on a sphere that do not qualify as "circular".