I am taking part in a robotics competition, where the challenge is to create a pair of robots which successfully navigate a series of obstacles. However, the rules state that of the two robots, only one must have a driving actuator. The other must somehow be moved by the other robot, WITHOUT PHYSICAL CONTACT.

I could think of either having sails on the non-driving robot, and moving it with fans on the driving one OR electromangnets on the driving one and permanent magnets with the opposite polarity on the non-driving one. However the problem with both is that efficiency falls off drastically with distance. Thus, I am looking for possible ways to overcome this problem.

Also, the driving robot has a cable power supply, while the non-driving one may only have batteries which can steer it.

I believe this question belongs to Physics as I want to know the actual forces possible, not their implementation in robots. Thanks :)
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The average *vector* momentum of an electron bound to an atom is exactly zero. (Otherwise, the electron would leave the atom!)

The average *magnitude* of the momentum can't be zero, because of the uncertainty principle. So Feynman is using the approximation $\vec p = \vec 0 + \Delta p \hat p$, where the magnitude $\Delta p$ comes from the uncertainty principle and the unit vector $\hat p$ points in a completely random direction.

As for your second question, you're almost there. The kinetic energy *does* become larger for an electron nearer the nucleus — and, thanks to the uncertainty principle, so does the momentum! It has to be this way because the kinetic energy is approximately $T=p^2/2m.$
This might be a bit helpful. 

The uncertainty in the momentum of an electron in an atom is defined as: $(\Delta P )^{2}= \langle P^{2}\rangle-\langle P\rangle^{2}$. An electron bounded by the nucleus, has an average momentum of zero, which means ($\Delta P)^2 = {\langle P^{2}\rangle}$, but again, Feynman's explain, as explained in the comment, was a really hand-wavy approach towards approximating the size of an atom.