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, $\phi$ = $\omega$t and $\vartheta$ = $\pi$ / 2 [1 + $\frac{1}{4}$ cos (4$\omega$t).

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$\omega$t) + $\frac{\pi^{2}}{4}$ sin$^{2}$ 4$\omega$t] $^{1/2}$

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}$$_{\phi}$b$\omega$cos [$\frac{\pi}{8}$cos 4$\omega$t] - $\widehat{e}$$_{\vartheta}$b$\omega$ $\frac{\pi}{2}$sin (4$\omega$t)

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,

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

ehild