My lecturer gave us some exercises on Dynamics with explained solutions in case we get stuck. One of the exercises is this:
A particle of mass $\space m \space$ is attached to the end of a light inelastic string of length $\space a \space$, the other end being fixed.
Initially, the particle hangs freely vertically below the fixed end, and is given a horizontal velocity $\space u \space$. After the string has turned through an angle $\space \theta \space$, show that the tension in the string is:
$$ m \left[g \left (3 \cos{\theta} - 2 \right) + {{u^2}\over a} \right]$$
My lecturer handwrites the solutions. Here is a snippet of what he wrote and what confuses me:
Notice how he subbed the $\space \ddot{r} \space$ with velocity $\space u \space$, but also subbed $\space {d \dot{\theta} \over dt} \space$ with velocity $\space u \space$.
Also, I can't tell whether he wrote $\space u \space$ or $\space \dot{u} \space$. If it's the latter, isn't $\space u \space$ just a constant and not a velocity function of time? Hence making the act of dotting it pointless?
Sorry for the overload of questions, I'm really stuck. I much rather prefer typed notes to avoid issues like this.