Integrating a vector equation two times

Question

The question is: A vector $\overrightarrow{R}(t)$ is a function of variable t, integrate the following equation two times $\frac{\text{d}^{2}\overrightarrow{R}(t)}{\text{d}t^{2}}\cdot\overrightarrow{R}(t)+\frac{\text{d}\overrightarrow{R}(t)}{\text{d}t}\cdot\frac{\text{d}\overrightarrow{R}(t)}{\text{d}t}=0$ and prove that $|\overrightarrow{R}|=\sqrt{Ct+D}$ where C, D the integration constants.

So far I have managed only the first integration using the following: $\frac{\text{d}^{2}\overrightarrow{R}(t)}{\text{d}t^{2}}\cdot\overrightarrow{R}(t)+\frac{\text{d}\overrightarrow{R}(t)}{\text{d}t}\cdot\frac{\text{d}\overrightarrow{R}(t)}{\text{d}t}=0\Leftrightarrow\frac{\text{d}}{\text{d}t}(\frac{\text{d}\overrightarrow{R}(t)}{\text{d}t}\cdot\overrightarrow{R}(t))=0\Leftrightarrow\frac{\text{d}\overrightarrow{R}(t)}{\text{d}t}\cdot\overrightarrow{R}(t)=C$.

And I am stuck here, so any help will be more than accepted, thank you for your time.

Answer 1

For the other step, note that

$$ \frac{d}{dt} |\vec{R}(t)|^2 = \frac{d}{dt} \left( \vec{R}(t) \cdot \vec{R}(t) \right) = \frac{d}{dt} \vec{R}(t) \cdot \vec{R}(t) + \vec{R}(t) \cdot \frac{d}{dt} \vec{R}(t) = 2 \left( \frac{d}{dt} \vec{R}(t) \cdot \vec{R}(t) \right). $$

Answer 2

HINT:

Note that we can write

$$\begin{align} \frac{d^2\vec R(t)}{dt^2}\cdot \vec R(t)+\frac{d\vec R(t)}{dt}\cdot \frac{d\vec R(t)}{dt}&=\frac{d}{dt}\left(\vec R(t)\cdot \frac{d\vec R(t)}{dt}\right)\\\\ &=\frac12\frac{d^2\,\left|\vec R(t)\right|^2}{dt^2} \end{align}$$