The whole thing is very, very simple and quite impossible.

Think first what would happen with manageable forces, such as masses of a few grams in collision.  (If you can't answer those, what is the point of nattering on about irresistible forces?  If you can't handle those, experiment on a Newton's Cradle till you get the picture; apart from anything else it should show you why the idea of immovable objects is nonsensical.)

Now, having got that clear, try the same style of thought on masses ten, or ten-thousand times bigger.

How do your answers differ? If you find that they differ in any way other than the sizes of the values, return to GO and start over; you haven't even found out what you are talking about, let alone what you ask.

But I am sure that you in fact find a deep truth: to counter a force of momentum X, you need a counter-force of at least X.

This does not require any deep maths, hm?

Not only does X + (-X) = 0 but X + (-X +D) = D.

We can tell in principle that if there is a Discrepancy, it will show in the result, and what shows in the result will be neither larger nor smaller than the Discrepancy. (There are practical complications, but they do not affect the principle.)

Understand?

Sure?

Well then what values of X and -X would you use for your irresistible forces? (And if you say "infinity" you will immediately be sent back to GO without 200 pounds, because you not only failed to do your homework on forces, but on infinities!)

Simple, wasn't it?

an 'irresistable force' means that there is nothing in the universe that can stop it.

an unmovable object is exactly such an object!

presto! it's a trick question.

it's like asking 'what happens when a square is round?' 

it's the kind of thing ppl would tell u - annoyingly - it's not a valid question.

There is a theological equivalent, intended to demonstrate that God cannot be omnipotent.

"Can God make a stone so heavy that He cannot move it?"

This is really a question of semantics rather than Physics.

 If there is such a thing as an irresistible force, then it must be able to accelerate every object, no matter how massive. Similarly, if there is such a thing as an immovable object, then it must be capable of resisting every force, no matter how big the force. Both of these postulates cannot be true at the same time. Therefore it is not possible for an irresisible force to meet an immovable object.

The physicists are right that this is all impossible and not worth worrying about, so let's cross over to the engineers and ask them! They will tell you that [1] the immoveable object will just be ground to dust at the localised point of impact. The bonds that hold the matter of the 'object' together just cannot hold out against the 'force'. Its called Hertz contact strees, and can be calulated. [2] If the force is carried by another impacting object, then it is in for trouble too: it will also turn into dust at its tip. Every time you drill through concrete you get this happening: the concrete breaks up into dust, and the impacting drill bit wears down too.

If the force is not carried by an object, then it's a field, and that field has to be generated by some apparatus [or in the case of gravity, by a massy object], so that apparatus will have forces propagated back into it, giving it anchoring troubles of its own in the opposite direction [think Newton's law of action and reaction here].

In some ways, every time you put a glass of water on the table you encounter an unstoppable gravitational force meeting an immovable object [table].In this case there is no grinding, because the table is strong enough [though it does deflect microscopically]. Gravity is an irresistible force [cannot be shielded], so in this case what happens to the table depends on how strong the material is: the impactor might cut its way through [like a diamond being pressed into wax], or just shatter the table outright [like a hammer smashing a diamond]. So the material strength comes into it in practice: what the engineers call its yield strength.

Now, if you add 'unbreakable' to the original problem, then I acknowledge there is a bigger problem. But the engineers would solve that too: they would tell you to Get real, Bugger off, and Take your problem to the philosophers!