[Math] Tetris-like falling sticky disks

Question

Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, if the disk center reaches $y=0$, the disk stops
with its center resting on the $x$-axis.
Here is an example of 1000 disks falling at uniformly random
$x$-locations within $[-50,50]$:

         

There are many questions one could ask about this (to me)
beautiful and intriguing structure (e.g., about its contact graph),
but to be specific,
let me concentrate on one quantity: the maximum height
$h_{\max}$ as a function of the number of disks $n$ and
the $x$-range $R$. (In the above example, $R=100$ and $h_{\max}=94.9$.)
It appears that $h_{\max}$ grows linearly, with $h_{\max} \approx n \frac{10}{R}$.
Here is plot, where each point is an average of ten trials:

                   

Two questions:

Q1. Is there a simple explanation of the growth of $h_{\max}$?

Q2. Has this process, or something close to it, been studied before?

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Ultimately I am interested in determining packing densities of
randomly jostled shapes, as explored in the earlier MO question
"Average degree of contact graph for balls in a box."
Sticky disks are a very simple model along these lines.

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Update (3Mar16).
An article by Ivan Corwin on KPZ universality has just appeared
(AMS Notices PDF),
including this figure to illustrate the "random ballistic" model:

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Users ansobol and Nechaev and Jeremy Voltz
previously pointed to the relevance of KPZ universality.

Answer 1

A discrete space version of your picture (with squares instead of circles) has been studied quite a bit, known as Ballistic Deposition. Here is a video of the process in action:

Ballistic Deposition on YouTube

This process is believed to be in the KPZ Universality class, so that the scaling limit of the height function can be described by the Airy Process with $t^{1/3},t^{2/3}$ scaling parameters.

Here is a short introduction to the theory:

KPZ Universality Class