So this might sound like a physics problem, but I am only putting it in physical terms so I can express myself better and make the whole thing clearer.
Suppose I shoot a bullet into a jelly. The deeper the jelly, the thicker it is. But this is a very special type of jelly, which ages very rapidly and so, as the bullet passes through it, the jelly thickens all around in a proportional manner with its density at every point on it.
So, in essence, this problem can be reduced to one dimension.
Now, why I am posting this question in the math section is because I am interested in how to tackle mathematically such an evolving problem. I am quite sure an integral must be used, but I do not see how.
So… to sum it all up: $v_0$ - initial velocity of the bullet when it first touches the jelly; $d(x)$ - the density of the jelly depending on its depth; $a(t)$ - the aging process of the jelly, which depends on time and is applied as: $a(t)*d(x)$.
My question is, in fact, a very simplified version of a charged particle in a magnetic field problem, so this is why I am asking it in a mathematical and general manner, so I can better understand the mathematical mechanism behind it. That is, please to whomever sees this as being interesting, please do not simplify it.