Can rapid acceleration have its own "version" of momentum? I'm a high school student and I've only taken AP Physics one.
Objects have momentum which is determined by their mass and velocity, but I'm wondering if there is something which continues increasing the velocity as the net force instantaneously changes from a very high number to zero (determined by its mass and previous acceleration). Let me give a scenario:
An object with 1 kg of mass starts at rest and begins accelerating at a very high acceleration, let's say 100,000 m/s^2. After one second exactly, all in one instant, the force which has been accelerating this object stops accelerating it. The velocity is now 100,000 m/s. Does the object still accelerate in the moments following the 2nd change in acceleration? Is there experimental evidence about this question? I would imagine, if this were true, that the effects would be very small and only apply for a very short time. 
 A: Nope. The reason momentum is conserved (at this level) is because Newton's second law, $\vec{F} = m\vec{a}$, basically says that force is the rate of change of momentum. If it weren't $\vec{a}$ on the right hand side but some other quantity, like jerk (the rate of change of acceleration), then mass times acceleration would be conserved like momentum (mass times velocity). Because Newton's second law is the way it is, though, forces (and therefore accelerations) can change instantaneously by as much as we want, in principle. In reality things are more fuzzy and smooth, but that isn't caused by some sort of conservation of acceleration, but instead the fuzziness of atoms.
A: No.  The change in momentum is given exactly as
$$\frac{d \mathbf p}{dt} = \mathbf F(t)$$ 
within the Newtonian physics framework.  When $\mathbf F(t) = 0$, so is the change in momentum, according to this equation.  More sophisticated (more accurate) theories do exist in extreme regimes of the very small and the very energetic (Quantum Field Theory, General Relativity), but these are called for at rather 'extreme' scales.
A: As the other answers have pointed out, there is nothing like momentum which would prevent instantaneous changes in acceleration within the laws of motion.  However, I would like to point out that in practical scenarios, it is often not feasible to have a force suddenly "vanish."  Forces caused by real effects do take some time to diminish.
This is not to say that there is a momentum like effect for acceleration as there is for velocity.  If you are calculating the acceleration of an object whose accelerating force instantaneously ceases,  its acceleration ceases as well.  However, you will find that in real life, such instantaneous effects do not exist.
As a real life example, many variable thrust rocket engines cannot respond instantaneously to commands to change their thrust.  This is because the thrust produced is related to several factors inside the rocket engine (such as the momentum of turbine blades or pintles).  At a "medium fidelity" level of modeling, one might choose to model the rocket as though it had a momentum-like effect on acceleration.  However, this would be a modeling assumption, not a law of physics.  It would be an artifact of simplifying the way a complex machine moves and behaves into a small set of equations.  Momentum, in the Newtonian sense, is much more fundamental than that.
A: I'll assume your experiment is far out in space where gravity or other forces cannot affect the experiment. Second I think your missing information about the force. You can't just say so many meters per second for one second unless you mean that's how fast it will be moving after one second of applied force. Either way there will not be acceleration (positive or negative) as soon as all forces are taken away. 
