The track curve of a particle is described with respect to an inertial cartesian coordinate system with the position vector with constants $a$, $b$, $\omega $ and time variable $t$.
$\vec{r}=acos(\omega t) \vec{e_{x} }+bsin(\omega t) \vec{e_{y} } $
i) With which dimensions are the constants $a$, $b$ and $\omega $ associated? Which form has the track curve (sketch)?
ii) Calculate velocity of the particle (vector) and their magnitude?
iii) Give the unit tangenten vector of the track curve.
What I think:
a) I dont know this part, not even sure what question really means.
b) I think i should use this formula $\vec{v}=\frac{d\vec{r}}{dt} $
c) I should only divide vector with magnitude from b): $T=\frac{\vec{r} ' }{| \vec{r}' | } $
Can someone help me with a) and tell me are b) and c) good?
To address part (i), the dimensions of $a$ and $b$ are length. These parameters denote the major and minor axes of the elliptical track. To draw such a track, the track would be elliptical, like an oval, and the "long edge" could be $a$, say, and the "short edge" could be $b$. Note that if $a = b$, then the track is circular since the position vector parametrizes a circle of radius $r = a = b$. In general, $a\ne b,$ and the track must be drawn to show that radius is not uniform and is instead scaled by a factor $a$ in the $x$-direction and $b$ in the $y$-direction. The parameter $t$ has dimensions of time. The parameter $\omega$ also has dimensions of inverse time, not only because it is an angular speed but also because the argument of the trigonometric functions must be dimensionless.