Derivating a velocity's components

[soopo]

Homework Statement

What is the derivate of the following equation?

$$v(t) = \dot{r}\hat{r} + r \dot{\hat{r}} = \dot{r} \hat{r} + r \dot{\delta} \hat{e}$$

where e is dependent on the angle, delta.

The Attempt at a Solution

My answer:

We know that
$$\hat{r} = \frac {r} {|r|}$$

=> $$\hat{r} = \frac {r} {|r|}$$

=> $$D( \dot{r} \hat{r} ) = \ddot{r} \hat{r} + \dot{r} \dot{ \hat{r}}$$
=> $$D( r \dot{\hat{r}} ) = \dot{r} \dot{\hat{r}} + \frac {r \ddot{r} } {|r|}$$

so we get

$$a(t) = \hat{r} ( \ddot{r} ) + 2 \dot{r} \dot{ \hat{r}} + \frac {r \ddot{r}} {|r|}$$

which is wrong.

The correct answer is

$$\dot{v(t)} = (\ddot{r} - r \dot{\delta}^2) \hat{r} + (r \ddot{\delta} + 2\dot{r} \dot{\delta}) \hat{e}$$

 

[foxjwill]

Are you working with vectors or scalars?

 

[soopo]

Are you working with vectors or scalars?

The derivate of a vector is a vector.
It consists of unit vectors which means that there is too scalars in the unit vectors.
So I am working with vectors.