Quandries...

A challenge was issued. Battle lines were drawn, but tonight a winner will be decided. In an all out battle royale we have in the blue corner, The Unstoppable Force! His foe, the undefeated adversary, in the red corner is the challenger, The Immovable Object. Who will win? Stay tuned as they proceed to duke it out. But first, some definitions of terms. As we know, force is equal to mass times acceleration. F=Ma. If we break out the calculus this turns into F=mdv/dt or the change in velocity over the change in time. An unstoppable object translates into an object with infinite force. Meanwhile though, we have the immotile object meaning it has an infinite amount of inertia. Inertia is a property of mass, as such momentum is a decent approximation of said inertness. Where, momentum is defined as the mass times the velocity P=mV. Ah, two infinities colliding to produce any of several different possibilities: If the force is the larger of the two countable infinities then the wall will move; if however, the wall has a greater amount of inertia than the force can act upon then the force will not only stop, but it will change directions as well. It is possible that both could annihilate each other or that they will equal each other, in that the immovable wall will not move nor will the force cease it will simply have encountered its equal. The force may simply change directions going around the wall, or possibly through it. DING! And there’s the bell, our prefight match up has ended and the two have collided (To officially answer the question we will need to know about the nature of the collision, is it completely elastic, entirely inelastic or a combination of something in between, but fear not for I have some tricks up my sleeve yet.)! What does it mean for a wall to be immovable since we are measuring its momentum it must have a velocity of 0, otherwise it would be moving, but if it has infinite momentum it must have an infinite mass(we’ll play with the prerequisite calculus necessary to deal with 0x∞ in a bit.). To be unstoppable what must an unstoppable force have? It must have either/ both an infinite mass or an infinite acceleration. If it has an infinite mass and at least some acceleration then one would suspect that because it has two terms whereas the wall has only one that the force should trump the wall, however, this is not the nature of infinities. Though it is true that one infinity can be much larger than another (You should read Cantor and his seating paradox to really get this one.), it may not be true that just because force has two infinite terms while momentum has but one, that the force is stronger. The other alternative is interesting though, if instead of infinite mass, the unstoppable force has an infinite acceleration this means it must change velocity in zero seconds which means it is not really in this universe since time and distance are related if the force exists outside of time it exists out side of this universe—which has interesting side applications for religion, but we shan’t venture into these just yet. So what do we do? We collide the force with the momentum. F/dv/dt=m è P= (F/dv/dt)V. We have solved force for the mass and replaced the momentum term with said force. Hmmm, it appears as though masses are not the solution to our problem, as in this instance they have both canceled. For kicks let’s see what happens if we solve for velocity. If you haven’t had calculus, panic! If you have, no sweat we’ll just separate the equation and we find we are left with dV/V=(F/P)dt. If you haven’t had training in the tools of Leibnitz and Newton, what we are going to next is smash the thing to itty-bitty tinsy tiny parts and then add them all back together. When we do this we are left with LnV=(F/P)T +C. Where C is our initial velocity, but since the wall by definition was not moving we are safe to say c=0 Taking the inverse log we are left with V=e^(F/P)t which in and of its self is pretty cool. It means that given time force and momentum you can solve for velocity. But let us continue our sadism one step further let us instead solve for position, recalling that v=dx/dt è dx= e^(F/P)tdt. Pulling our favorite trick of integrating the crap out of it again (for the non math geeks who have actually stuck with it this far integrating is the trick we pulled where we smashed it into bits and added up all of the broken bits), we are left with x = (P/F)e^(F/P)t+C Hmmm, that pesky C popped in again, but fear not, for we are crafty theoreticians and are prepared for just such an unfortunate event. Based on careful prior planning we have placed our wall exactly on a spot we have called 0. Ho, ho, luck is with us for c=0 yet again. Woot! (It is important to note that in this instance C refers not to the speed of light, but to some unknown constant. Letting it equal 0 is perfectly acceptable for our purposes.) Ok, now for the fun stuff. Let’s see what happens when we let both force and momentum go off to infinity. If x goes negative the force wins as the wall will have been pushed back. If x goes positive it means the wall will force back the force. If it stays zero, its hard to tell whether the force pushes the wall equally, or if both are destroyed. If x goes to infinity the force won. And now for the moment of truth, we let both F and P limit to infinity and what do we get? (We should also note that the force and the wall are opposing each other and should therefore be opposite in sign from each other.) ( -∞/∞)e^(∞/∞)t . This is clearly undefined, resume panicking! Oh yeah, we have L’Hospital’s Rule e^-∞ = 0 so if P/F goes to infinity the value is still undefined, if it goes to any finite value then its still 0. So dP/dF= -1/1=-1,- 1x0=0 The wall wins! The unstoppable force has been either stopped or destroyed. The wall having only mass to go on beat the multiple valued unstoppable force! Though, sadly we know the position of the wall so well we can longer be certain if it has a momentum, it may have destroyed itself stopping the force, though Ladies and Gentlemen, it has been done, the force has been stopped.