Ooze bears


Image source: Critical Hits
Oozebears, semi gelatinous bear-like creatures, exist in bear society to recycle waste and effluents, and thus are aptly described as 'cesspit dwellers'. Their function in the invasion of earth is to serve as infiltrators:

Because they are literally cesspit dwellers, they were perfectly suited to infiltrating the sewers of every city in the world, lurking in fetid lagoons and oozing through pipes. They have even infiltrated isolated black water tanks in peoples' homes. When the time comes, they will emerge from the pipes and attempt to severely disrupt human society by indiscriminately killing.

One thing of note about ooze bears is that they are really significantly less intimidating than an actual bear, in the sense that they are much less capable of killing a person in a given period of time. An ooze bear kills by partially absorbing its victim, and excreting a large volume of acid, and while the damage done by this attack is significant, it still takes a matter of approximately five minutes to kill a person.

The ooze bear, though technically gelatinous and thus a solid, reacts to trauma on a cellular level by disassociating into a fluid, and flowing onto the ground to reform. When this damage becomes too extensive, a chain reaction occurs which causes the whole ooze bear to turn into a fluid, and after this it cannot reform and becomes mostly harmless. It is has been my observation that once an ooze bear loses about forty percent of its mass as fluid-loss, it dies in an amount of time equal to the following equation:

t=[1/(1.6-4*%)]

where t is the time to death in minutes, and % is the fractional percentage of mass disrupted. Typically, ooze bears take up no more than two cubic meters of space, so this equation can be re-written as

t=[1/(1.6-4*(V/2))]

And, of course, these equations are held to the restriction that %>.4. And from that, you see that so long as total damage crosses this threshold, the ooze will probably lose integrity well before killing, although more damage is undoubtedly preferable.
So, in order to kill the ooze bear, we know that we only really need to know the diameter of a projectile, and how
far that projectile penetrates into the ooze bear. Because we know that an ooze bear disassociates into a fluid in reaction
to trauma, the distance a projectile goes through an ooze bear can be calculated by how far it can travel through a fluid
of the given characteristics. For that, we start with the equation dictating force of drag, easily found in our textbook
on page 145.

F=.5*C*p*A*(v^2)

where C=the drag coefficient, p=material density, A=the cross section of the projectile, and v=velocity.

To determine the penetration of the projectile, we consider its kinetic energy, or rather the change in kinetic energy,
which is known to be force times distance, or, in this case:

dE=-.5*C*p*A*(v^2)*dx

and of course the sign changes here because the change in energy of our projectile will be negative.

The equation we are left with brings us closer to a productive equation, but velocity is problematic. It can be said,
however, that v^2=(2*E)/m, from our understanding of kinetic energy, and the equation again changes to

dE=-C*(E/m)*p*A*dx

This equation is now very close to being useful, except that it needs to be integrated, and the value of energy consolidated,
which all sounds very tedious except that this is the differential equation for exponential decay, written, in this case, as

E=E0*e^(r*x)

where r=-C*A*p*(1/m), the constant of our equation, E0 is initial energy, and x is of course distance traveled. To prove this
we rewrite dE as

dE=r*dx*E

then substitute in E0*e^(r*x) for E

dE=r*dx*E0*e^(r*x)

Which we easily see is definitely the derivative of our energy equation, in the general form of exponential decay! Now we re-write this equation in its expanded form, because it is important and useful for us to know which constants we need to define.

E=E0*e^(-C*A*p*(1/m)*x)

And we also note that we now have the following list of constants: E0=initial energy, C=the drag coefficient, p=the material density, A=the cross section of the projectile, m=projectile mass, and finally x=distance traveled.

Note that by the equation we have described, there is technically no limit on the theoretical penetration of a round when the behavior of the body it is passing through is hydrodynamic. Of course, there is a limit past which round penetration can no longer be calculated using hydrodynamic performance, and in this case that limit is when the pressure exerted by the round onto the oozebear is less than F/A=50,000Pa

And, with all of this done, you can finally begin to analyze the performance of several different munitions, as since the only practical damage we care about is the volume of liquefied oozebear any further trajectory that can be mapped of our projectiles is irrelevant.

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