Consider two identical objects of equal mass collide with eachother with equal opposing accelerations. Wouldn't the forces of the objects cancel each other resulting in the objects both comming to a stop? If this is true how can a collision of this nature be elastic if the forces always cancel and the objects come to a stop?
Take the concept of conservation of momentum.Here, it means that the momentum of the two objects remains constant before and after collision.
m1*v1 + m2*v2 =const
where m1, v1 and m2, v2 are the masses and velocities of the two objects in consideration.
You have stated that the two of them have equal masses and velocities, with the velocities in the opposite directions.
i.e. m1=m2=m and v2= -v1
initial momentum= mv1 + m(-v1) = mv1 - mv1 =0
so, conservation of momentum states that the final momentum will also be zero.
Elastic collision : If the objects collide perfectly elastically, then both will retrace the path which they came in. Since they have equal masses, by symmetry, they will have equal velocities.
thus again, final momentum of the two objects is
m*(-v1) + m*v1 = 0
Inelastic collision :If the collision is perfectly inelastic (putty is very good for this experiment), they stick together and by symmetry (since they had equal masses and velocities) they would come to rest at the place of collision rather than moving along one of the two direction. (if they did move along a path, it would mean than one of the objects had a higher momentum than the other, which is not the case here.)
So, final momentum here is zero because of the zero velocity of the compound object so formed.
Now, in the question, you should have been clear as to whether the objects had uniform velocity or if they were accelerated.
If they were accelerated, and both by equal forces in opposite directions, such as to increase their relative velocity, use F=ma and v=u+at to obtain the velocity at the instant before the collision.
Even then, the conservation of momentum will hold as it is simply like using a different value for "v"
Wouldn't the forces of the objects cancel each other resulting in the objects both coming to a stop?
Kinematics is solved simply by using the conservation laws . In this case conservation of momentum and conservation of energy are active, as you have not introduced angular momentum.
Let us consider the instantaneous impact. The momenta are equal and opposite so they add to zero, the energy is equal and of the same sign so it will add up to twice the energy of one of them. Conservation of energy requires that the energy should go someplace. In an elastic collision this is taken up by each mass going off with an opposite momentum and an equal energy.
This is the elastic case.
If this is true how can a collision of this nature be elastic if the forces always cancel and the objects come to a stop?
They only come to an instantaneous stop and bounce off due to the conservation of energy, unless the objects are breakable or can be deformed. In this case, they may be deformed and the energy is taken up (conserved) as heat , they may break up into pieces setting up another problem of conservation of energies and momenta, or do both.
Momentum and energy have to be conserved. Conservation laws are the result of innumerable observations and they have to hold.
Your mistake is to think that the forces cancel.
You are quite correct in thinking that when you have two objects of equal mass colliding there are two equal in magnitude and opposite in direction forces acting on the two objects. That is Newton's third law in action. However those two forces are acting on different objects. So one object has one of those forces acting on it causing that object to accelerate whilst the other object has the other force acting on it causing that object to accelerate.
In the case of your example because the two masses of the two objects are the same the magnitudes of the accelerations will be the same but opposite in direction.
What happens during and after the collision depends on the objects but during the collision the kinetic energy that the object had must be converted into other forms of energy because there will be a time when both objects are not moving. The kinetic energy is stored as elastic potential energy - think of the bonds between molecules being little springs and also converted to heat and sound and permanent deformation of the objects in which case the bonds (springs) between the molecules are permanently broken.
In an elastic collision all the energy is stored as elastic potential energy and is recoverable so that after the collision the kinetic energy of the objects is the same as before. The other extreme is for the objects to stick together after the collision when a lot of the kinetic energy the objects had is converted into heat, sound and permanent deformation. Obviously there are a whole range of possible collisions between these two extremes.
Because the two forces acting on the objects are equal and opposite and occur for the same time they must apply equal but opposite impulses (forces $\times$ time) on the two objects and this results in the change in momentum of the two objects being equal and opposite.
Looking at the two objects together as a system those forces the objects experience during the collision are internal forces and do not count as external forces. If there are no external forces acting on the two objects then their total momentum will not change - the law of conservation of momentum - however the momentum of one of the objects can change provided the other object has an equal but opposite change in momentum.