# Determining the velocity function

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1. Dec 5, 2016

### doktorwho

1. The problem statement, all variables and given/known data
Given the $r(t) = ae^{kt}$ , $θ(t)=kt$ find the velocity function that is dependent on $r$.
$v(r)=?$
2. Relevant equations
3. The attempt at a solution

My attempt:
1)$r(t) = ae^{kt}$
2)${\dot r(t)} = ake^{kt}$
From the first equation:
$\ln {\frac{r(t)}{a}}=\ln e^{kt}$
$\ln {\frac{r(t)}{a}}=kt$
$t=\frac{\ln {\frac{r(t)}{a}}}{k}$
Replacing the $t$ in the second equation i get:
${\dot r}=akr$
Shouldn't this be the answer? In the answers it says ${\dot r}=\sqrt2r$?

Last edited: Dec 5, 2016
2. Dec 5, 2016

### Staff: Mentor

Your original equations are incorrect. $\theta$ is not a vectorl where are your unit vetors i these equations?

3. Dec 5, 2016

### doktorwho

Yeah, no vectors, just the parametric equations of motion given. So whats wrong now?

4. Dec 5, 2016

### Staff: Mentor

If you are going to determine the velocity vector, you need to start out by expressing the position vector as $\vec{r}=r\vec{i}_r(\theta)$ and taking into account the fact that $\vec{i}_r$ is a function of $\theta$, that $\theta$ is a function of time, and that derivative of $\vec{i}_r$ with respect to $\theta$ can be expressed in terms of $\vec{i}_{\theta}$.

5. Dec 5, 2016

### doktorwho

So the polar coordinate,
$\vec r(t)=ae^{kt}\vec e_r$
$θ=kt$
$\vec v(t)=\dot r\vec e_r + r\dot θ\vec e_θ$
$\vec v(t)=ake^{kt}\vec e_r + ae^{kt}k\vec e_θ$
$v(r)=\sqrt2r$
This should be it.