If two "ships" are pulling on the opposite ends of a pole 22,353,840 miles long with equal force how long will it take for the second ship to begin moving if the first ship is disconnected at the point of connection with the pole?

Andrew & Steve are correct. For sake of simplicity, lets assume the whole system to behave linearly. It will help to get some qualitative assessment of the problem. Let the pole to be linearly elastic with coefficient of elasticity E, area of cross section = A and mass density m. For the pole to be perfectly rigid, we can always take that E tends to infinity. (Why that is not possible, we will come back later). Also assume the pole to be a perfect mechanical column (unidirectional tension and compression bearing member) When the tension in the pole is is in static equilibrium with value of tension being T, the equilibrium stress developed in it at any point x along the length is given by S such that: T = S*A = E*A*dw/dx (w being measured on stretched pole and x being measured on unstretched one) . Now a disturbance in static equilibrium is added by changing T by small amount. The problem is no longer a static one but enters the domain of dynamics. Now S and T are no longer in equilibrium at the moment of introduction of the disturbance. But S will try to change in order to get back to equilibrium. The equation of motion of the small element at x now becomes: m*A*dx*(d^2w/dt^2) = T+dT - S = dT => m*A*(d^2w/dt^2) = dT/dx => d^2w/dt^2 = (E/m)*d^2w/dx^2 which is a wave equation giving the speed of propagation (same as speed of sound) = \sqrt(E/m). This is just a linear analysis. In real case, it would involve some order of non-linearity. Hence, the "information" of disconnect will propagate with speed of sound. Now, if the pole is infinitely rigid, the speed of propagation would be infinite. But that would violate Einstein's SR. But in real world, such value of E is not possible. After all, E comes from atomic configuration material which  in turn is governed by Maxwell equations. The specific elasticity E/m is always upper-bound by c^2.

's answer looks correct. Assuming that the pole is a solid steel rod, the speed of sound will be around 5000 m/s. Converting the length of the rod to meters and dividing by this speed, and then converting the answer (around 36 million seconds) into days, you get 83.275 days, which is Andrew's figure. This seems a surprisingly long time to me, from an intuitive viewpoint, but then physics sometimes can be counterintuitive, and you must go with the maths, not your instincts. Of course, the problem posed by the question is far too idealized and abstract to yield a realistic answer, anyway. There is no material that is 'perfectly rigid'; even a rod made from a single crystal of pure diamond would stretch, bend, deform and transmit vibrations - especially one 22 million miles long! And can you imagine how much mass such a rod would have? Interestingly, if you replaced the steel (or diamond) rod with a carbon nanotube, you can get the time down to around 15 days, due to the very high speed of sound in carbon nanotubes (the value I found via a quick Google is 27,000 m/s).

Lets assume that the pole is not perfectly rigid, but at least simply that it is infinitely strong and very elastic (as in energy input as stretch is returned close to 100%) The pole has stretched a teeny weeny bit, each atom in the pole is slightly farther away from its neighbours than before. Now the beauty is that atomic forces caused by electromagnetic repulsion (why u cant push one atom into another is because the electrons repel) also travel at the speed of light at most .... So as soon as one end is cut, the very first layer of atoms move towards their neighbors, but repulsion is not instantaneous - it happens through the exchange of what are called "virtual photons". When one particle pulls or pushes another by any of the 4 forces in the universe, force carrying particles are exchanged. For electro-magnetic interactions, photons are exchanged. Thus even in an "ideally rigid" material, the information of the separation of the spaceship will only reach at light speed at most. In a similar vein, if the sun vanished now, the earth would still orbit in an ellipse for 8 minutes and then it would shoot off like a slingshot in a straight path. This is simply because "now" on the sun is not "now" for us. It is only "now" after 8 minutes, because any event that happens on the sun takes 8 minutes to affect us at the very least.

According to me the ships are at a state of static equilibrium when their forces are equal and opposite. And when the ship is disconnected from the pole the state of equilibrium is broken. Now since the load is continuously acting on the two ships they must start motion instantaneously. i.e no force is transmitted from the pole onto the ships to cause them to move. Therefore a question of time delay do not occur at all.

Im a biologist not a physics person. However, my thought is to divide by c(speed of light). Not for the reason above, but because csusality or cause and effect propagate at that speed. Thrrefore the fact of the disconnection of the ship would have meaning at that distance in d/c or, off the top of my head about two minutes or so.

The speed of sound in the pole. There can't be any such thing as a perfectly rigid pole, because the speed of sound cannot be infinite.

if i understand correctly, these three units, ship-pole-ship configuration is acting as a single unit. its not interlinked like a train i.e. there are no couplings. now if an object is being pulled apart, and is cut or broken into two, the new created distinct units should immediately move away in respective directions. specially here when no other force (gravity, friction, etc) are absent.

Rigid poles cannot exist in our Universe. All else is commentary.