Some issues concerning a plot created as a `tikzpicture`

Question

The above tikzpictureplot was created mainly adapting the code of Create xkcd style diagram in TeX . I located the dots and labels by trial and error. What are the better options? To create the entire plot should I use draw commands instead?

Because I noticed that xtick={},ytick={} did not remove the axes tick marks, I put two marks only in each axis, hiding them by points O, S and R. Is there another better method?

As you can see the blue curve y=f(x) is not totally smooth as it should be. I thought that might be due to inaccuracies when evaluating the square root of small numbers. Can one overcome this issue?

Here is the code I used:

\documentclass{standalone}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{pgfplots} 
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\pgfplotsset{every axis/.append style={line width=0.9pt}}
\begin{axis}[
    axis x line=middle,  
    axis y line=middle,   
    ymax=2.2, ylabel=$y$, 
    xlabel=$x$,
    xtick={0,3},
    ytick={0,2},
    xticklabels={},
    yticklabels={}
    ]
  \addplot[domain=0:3, blue, thick]    {sqrt(x^3/27)-sqrt(3*x)+2}; 
  \addplot[domain=0:1.5, red, thick]  {-(x/4+7/8)*sqrt(2)+2}; 
  \addplot[domain=0:3.2, black, thin] {0};
\end{axis}

\node at (-0.25,-0.25) {$O$};
\node at (3.4,0.9) {$P(x,y)$};
\node at (-0.7,2) {$Q(0,ut)$};
\node at (-0.7,5.1) {$R(0,uT)$};
\node at (6.7,-0.3) {$S(L,0)$}; 
\node at (2.4,1.8) {$y=f(x)$};
\node at (4.8,0.4) {$s=vt$};

\fill(0,0) circle (2pt);
\fill[blue](0,5.15) circle (2pt);
\fill[blue](3.2,0.6) circle (2pt);
\fill(0,2) circle (2pt);
\fill[blue] (6.4,0) circle (2pt);
\end{tikzpicture}

\end{document}

EDIT: To add dots at the origin and Q I added to percusse's code

\addplot [only marks,mark=*] coordinates { (0,0) (0,0.76) };
\node[below left] at (axis cs:0,0) {$O$}; 

before \end{axis}. The last line should type the origin, as commented by percusse, but for some reason unknown to me it doesn't work. The initial code of mine \node at (-0.25,-0.25) {$O$}; after \end{axis} does type O.

Answer 1

As you have suspected, pgfplots offers a tight integration to TikZ-like syntax. I've put a few different types of doing things so you can choose your own favorite.

First, the axis coordinate system is handier to place things and you can access to that cs by using (axis cs:xcoord,ycoord) as I did with P(x,y) node. Also just before placing an ending ; to the addplot command you can place nodes as if it is a TikZ path(it is actually but still in a nontrivial way). It's better to place the markers as an individual plot with supplied coordinates with only marks and then followed by the marker type etc. Lastly, I've removed every label and added custom tick labels via extra ticks extra tick labels keys.

\documentclass{standalone}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{pgfplots} 
\pgfplotsset{compat=1.7}

\begin{document}

\begin{tikzpicture}
\pgfplotsset{every axis/.append style={line width=0.9pt}}
\begin{axis}[
    axis x line=middle,
    axis y line=middle,
    ymax=2.2,xmax=3.2,
    tick style={draw=none},
    ylabel=$y$, xlabel=$x$,
    xticklabels=\empty,
    yticklabels=\empty,
    extra y ticks={0.76256,2},
    extra y tick labels={ ${Q(0,ut)}$ , ${R(0,uT)}$ },
    extra x ticks={3.0},
    extra x tick labels={ ${S(L,0)}$ },
    ]
  \addplot[domain=0:3, samples=100,blue, thick]  {sqrt(x^3/27)-sqrt(3*x)+2} 
        node[pos=0.3,above right] {$y=f(x)$} node[near end,above right] {$s=vt$}; 
  \addplot[domain=0:1.5, red, thick]  {-(x/4+7/8)*sqrt(2)+2}; 
  \addplot [only marks,mark=*,blue] coordinates { (0,2) (1.5,0.23) (3,0)};
  \node[above=2mm,blue] at (axis cs:1.5,0.23) {${P(x,y)}$};
\end{axis}

\end{tikzpicture}

\end{document}

Apologies for the 0.23, I've just guessed it visually.

EDIT: Also you can increase the sample number via samples=n key. See the relevant plot.