Do tectonic plates "float" over the mantle and "collide" like icebergs? I'm always hearing about tectonic plates as large chunks of crust floating on the mantle just like boats. In timescales of millions of years they move and even collide.

But I'm starting to think this is just a pop-science model, or a methaphor, for a much more complex situation. I didn't made the calculation but I feel that the mass of the Indian plate is not as large to account for the uplifting of the Himalayas just by kinetic energy transfer. At those speeds (puny, even if large compared to other plate movements) wouldn't the material stress of the Eurasian plate had absorbed all the momentum by now? I get that the inertia of India is huge, but is it really as much as to keep pushing even today? Does it has so much inertia that it is still slowing down as it pushes Eurasia? I feel like there is a constant force been applied tangentially to the surface of the plate that could account for this instead of just a freely moving plate smashing another like two icebergs in the artic sea.

Does the idea of plates floating like boats and the idea of them interacting by mechanical collisions is really somehting more than a suggestive way of viewing a process that takes enourmous amounts of time, energy and mass so that we, insignificant and ephimeral creatures, can have a toy model in our minds? Is India been pushed by a force or is it really just moving like a billiard ball until it collides with another and changes its momentum?



EDIT:
Ok, so I've actually done the calculation now:

According to the USGS, the Indian plate had a speed of $v =9 \; m/century = 2.85\cdot 10 ^{-9}\; m/s$.

The surface area of the plate is $A = 1.19\cdot 10^{7}\; km^2$. If we suppose the thickness of the crust here to be of $h = 50\; km$ (which is thicker than it probably is) then the volume of the plate is $V = A\cdot h= 5.95\cdot 10^{8}\; km^3 = 5.95\cdot 10^{17}\; m^3$. We can estimate the mass of the plate by assuming a density of $\rho = 3\; g/cm^3 = 3\cdot 10^3\; kg/m^3$ (this density is higher than the average we should expect for the crust so we are not been very conservative at all). Thus the mass of the Indian plate is around $m = \rho V = 1.79\cdot 10^{21}\; kg$ in the best case scenario.

Then the kinetic energy of the Indian plate had to be lower than $E_k = \frac{1}{2}mv^2 = 7269\; J = 1.7 \; cal$, which is even less than what @Keith McClary has suggested since this is less than a $1/300 \; th$ of the energy of a candy bar.

Now, we can use Newton's Second Law of Motion in the form $\Delta t = mv/F$, where $m$ is the mass of the Indian plate and $v$ is its velocity, to get the time needed to stop the plate, $\Delta t$, when we apply a constant force, $F$, against its motion. Even if the kinetic energy is insignificant it is not easily absorbed during a collision due to the huge inertia of the plate. But still if we suppose $1$ million people, each person pushing with $3000 \; N$ of force then those people could have stopped the continent in less than $\Delta t = 30\; minutes$. A single weight-lifter would have been able to stop the entire Indian continental plate if he pushed with $F = 8000\; N$ for about $\Delta t = 20.2\; years$.

I think that the mechanical stress of the entire eurasian continent creates larger forces that a single human and this "collision" has been going on for millions of years (not 20 years) and is still going on. So this is where it looks absurd to me to talk about a "collision" of plates for the formation of the Himalayas. The driving mechanism has to be a huge force pressing the Indian plate against Eurasia.
you are missing a big factor, the plates are not moving due to the momentum of an initial impulse. They are being actively moved by the push and pull of mantle convection. Much like how icebergs are pulled along by ocean currents. the iceberg analog however breaks down because icebergs melt before they can do much complex interaction, where as continental plate material is more or less permanent. also icebergs are subjected to far lower forces than continents, a closer analogy would be ice in glacier. 

The amount of energy the mantle is supplying is massive, far far more proportionally than an iceberg is subjected to, more than enough to drive continental plate together and build mountains.  if you are interested in the detailed mathematics you can start here. 

The continents float on the mantle becasue they are less dense, but they are moved by mantle movement transferring energy.