your way and then compare it to my way of thinking.
In your scenario the ping pong ball has velocity (thermal molecule) it collides with another molecule that has less kinetic energy, the collision imparts kinetic energy to the one that it collides with, and that in turn transfers it to the next one and so on. The result being that by the time the kinetic energy is distributed between all the molecules there is very little (if any) extra energy left to move the molecules at the end of the travel.
I agree with that scenario IF the molecule is not "radioactive", in other words if the molecule has only the kinetic energy imparted to it by the "gun" or the heat from the original source.
But the radioactivity changes the situation in a way that you are not taking into consideration.
Fission is a nuclear process in which a heavy nucleus splits into two smaller nuclei. An example of a fission reaction that was used in the first atomic bomb and is still used in nuclear reactors is <super>235</super>U + n ----> <super>134 </super>Xe + <super>100 </super>Sr + 2n
http://www.lbl.gov/abc/Basic.html
The speeding atoms of the radioactive molecule not only expend their kinetic (heat) energy but they "generate" energy from within through the process of "fission."
In the scenario with the speeding "radioactive" ping pong ball, the ball will explode at a certain point in its travel, in the process of explosion the ping pong ball (atom) releases more energy in the form of "shrapnel" (alpha, beta, and gamma energy) this energetic "shrapnel" then increases the kinetic energy of the surrounding atoms and molecules. (Nuclear reactors would not work without this effect.)
Not only that but some of the flying shrapnel pieces can have enough energy, not only to increase the kinetic energy of the surrounding balls, (heat the molecules) but also to "explode" some of the ones that get in the way, in other words, radioactive material that is in close proximity to non radioactive material causes the non radioactive material to become radioactive.
Now you are shooting many of these exploding bombs into the "room that is fifty miles long" and you are continuing that process year after year, eventually the "radioactivity" will reach the other end of the room, where they encounter other energies that are waiting there, such as cosmic rays, the combined energy then may be enough to cause the molecules to escape the gravitational field.