Finding the Orbit of a Particle with Varying Position Vector

[d2x]

I'm given the position vector as a function of time for a particle (b, c and ω are constants):

$\vec{r(t)} = \hat{x} b \cos(ωt) + \hat{y} c \sin(ωt)$

To obtain it's velocity i differentiate $\vec{r(t)}$ with respect to time and i obtain:

$\vec{v(t)} = -\hat{x} ωb \sin(ωt) + \hat{y} ωc \cos(ωt)$

Now i have to describe the orbit of this particle. I'm quite clear that if b=c the orbit is perfectly circular with constant tangential speed. But if b≠c (let's say b>c) is the motion elliptical with ±b as the semi-major axis?
Thanks.

 

[gneill]

I'm given the position vector as a function of time for a particle (b, c and ω are constants):

$\vec{r(t)} = \hat{x} b \cos(ωt) + \hat{y} c \sin(ωt)$

To obtain it's velocity i differentiate $\vec{r(t)}$ with respect to time and i obtain:

$\vec{v(t)} = -\hat{x} ωb \sin(ωt) + \hat{y} ωc \cos(ωt)$

Now i have to describe the orbit of this particle. I'm quite clear that if b=c the orbit is perfectly circular with constant tangential speed. But if b≠c (let's say b>c) is the motion elliptical with ±b as the semi-major axis?
Thanks.

Yes, the larger value will determine the semi-major axis, the smaller will determine the semi-minor axis of an elliptical trajectory. Your expression for r(t) is one form of the equation for an ellipse.