# circular(ish) motion?

how would one describe as a function of time the position vector of an object with initial position $$\vec{x_0}$$ and velocity $$\vec{ v_0}$$ experiencing a central force of constant magnitude around the origin.

The closest thing I have done to this is circular motion, however, all of it descriptions are based on the constraint that the object is moving in a circle, which would not, at least initially, be the case here.

• What is "centripetal force" supposed to mean if the object is not moving in a circle? – ACuriousMind Apr 8 '19 at 23:10
• the direction is always directed to the center as defined – user225234 Apr 8 '19 at 23:17
• If you just mean a force directed towards a fixed center, then that's a "central" force, not a centripetal force. – ACuriousMind Apr 8 '19 at 23:19
• I see, thank you for that correction. – user225234 Apr 8 '19 at 23:22
• the question has been changed – user225234 Apr 8 '19 at 23:23

An easy way is the following: take the velocity $$\vec{v}_{\mathrm{circ}}$$ that is equal to the velocity that the object would have if it were in uniform circular motion with the given force and its current position were $$\vec{x}_0$$.
Rewrite the velocity $$\vec{v}_0$$ as
$$\vec{v}_0 = (\vec{v}_0 - \vec{v}_{\mathrm{circ}}) + \vec{v}_{\mathrm{circ}}.$$
The motion of the object is then the circular motion associated with $$\vec{v}_{\mathrm{circ}}$$ and the given force plus the uniform straight motion $$(\vec{v}_0 - \vec{v}_{\mathrm{circ}})\,t$$.
To convince yourself that this works, substitute $$\vec{x}'(t) = \vec{x}(t) + \vec{v}'\,t$$ in the equation of motion, assuming that $$\vec{x}(t)$$ is a known solution of the equation.