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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.

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

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