# Derivating a velocity's components

1. Oct 24, 2009

### soopo

1. The problem statement, all variables and given/known data

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.

3. The attempt at a solution

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.

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

2. Oct 25, 2009

### foxjwill

Are you working with vectors or scalars?

3. Oct 25, 2009

### soopo

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.