Interesting.
Lets start by defining what "moving" means. Here essentially we have to differentiate between uniform motion (at constant velocity) vs accelerated motion.
UNIFORM MOTION: If you intent to move the wood at constant velocity then, technically you are not supplying any extra information than that's already there. You don't have to do anything it will keep moving at the same speed forever.
ACCELERATED MOTION
This is when things get very interesting.
FORCE REQUIRED -
For any amount of acceleration, you would require to apply force. This case an insanely huge amount of force. Issue with that? Well, every material has various range range of forces that it can handle. This force decides:
- whether the material will act as a rigid body OR
- like a Elastic body (like a spring - can completely reform back to the original state from any deformed states) OR
- like plastic (like a sponge - cant restore form deformation) OR
- PHASE TRASITIONS AND/OR MICELLANEOUS EFFECTS (if you compress charcoal enough you can get diamond out of it ,and other weird stuff characterized by the fact that "external energy supplied anticipating a movement actually goes into internal energy in a drastic way")
CASE 1: When Material is RIGID -
In Newtonian Mechanics, parts of a rigid body doesn't move wrt to each other, implying that the applied force instantaneously get to act on COM moving the whole body instantaneously. As you are aware, this is not quite right.
Incorporating Special relativity, RIGID body idea is extended to what is called as Born Rigid Body which in out case can be naively stated as information of the force travels at the speed of light. Hence there is a slight delay between action (applying force) and its effect (acceleration of body). The exact calculations are way too complicated so lets just say the delay in time is $T_1$.
CASE 2: When its behaves as an elastic body -
This means that when you apply force at a point, that portion initially get compressed and then the compression gets carried away (like the propagation of sound). In the linear/spring approximation, these compressions form uniform waves commonly called as acoustic waves and the corresponding wave velocity is the acoustic velocity is the speed at which the information travels. There are higher order(more better) approximations, but without going into details lets call the time delay in this case $T_2$.
CASE 3: Sponge -
If you have a huge sponge (huge in terms of size, but more importantly mass) and you jump into it... that's exactly what's gonna happen. The material is going to get compressed until it reaches close to the other end. And then CASE 4 takes place. The time delay $T_3$ is pretty high. $T_3 \to \infty $
CASE 4: is when strange things happen
This is when the compression force is way higher than inter-lattice repulsion and hence they start fusing into each other resulting in heavy departure from Classical Models and Quantum Phenomena starts to dominate. There are an hundreds of possibilities from here (not that I know all of them) and so its pretty pointless to go on further. Rough guess of Time delay is that $T_4$ can be anything greater than $T_3$ to $\infty$.
To Conclude -
The order of time delay depending on how much force you give is $T_1<T_2<T_3<T_4$. And to say further, practically (considering all the materials we know of) it would be highly unlikely to achieve anything other than CASE 3 and CASE 4 owing to the fact that to accelerate anything even close to the mass of sun would require tremendous amount of force to have any appreciable change in state of motion.
