Quick question with spherical coordinates and vectors

[Johnson]

So here's the question:
An ant crawls on the surface of a ball of radius b in such a manner that the ants motion is given in spherical coordinates by the equations:
r = b, \phi = \omegat and \vartheta = \pi / 2 [1 + \frac{1}{4} cos (4\omegat).

Find the speed as a function at time t and the radial acceleration of the ant.

I found the speed, doing \left|v\right| = b\omega[cos^{2}(\frac{\pi}{8}cos 4\omegat) + \frac{\pi^{2}}{4} sin^{2} 4\omegat] ^{1/2}

Now I don't even know where to begin to take the derivative of that, lol. I know i derive the actual vector v, not the magnitude of it. But how do i derive e_{\phi} and e_{\vartheta}?

I got for velocity

v = \widehat{e}_{\phi}b\omegacos [\frac{\pi}{8}cos 4\omegat] - \widehat{e}_{\vartheta}b\omega \frac{\pi}{2}sin (4\omegat)

Any help on deriving that to find acceleration would be awesome :s Maybe I'm missing a rule with \widehat{e}_{\phi}, but I'm getting stuck.

Thanks :)

 

[Johnson]

Any help would be greatly appreciated.

 

[ehild]

See: http://en.wikipedia.org/wiki/Spherical_coordinate_system,

"Kinematics"

In spherical coordinates the position of a point is written,

\mathbf{r} = r \mathbf{\hat r}

its velocity is then,

\mathbf{v} = \dot{r} \mathbf{\hat r} + r\,\dot\theta\,\boldsymbol{\hat\theta } + r\,\dot\varphi\,\sin\theta \mathbf{\boldsymbol{\hat \varphi}}

and its acceleration is,

\mathbf{a} = \left( \ddot{r} - r\,\dot\theta^2 - r\,\dot\varphi^2\sin^2\theta \right)\mathbf{\hat r} <br /> + \left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right) \boldsymbol{\hat\theta } <br /> + \left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \mathbf{\boldsymbol{\hat \varphi}}

ehild