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In the case where colliding objects move at constant velocity it is standard and very convenient to use conservation of momentum and conservation of energy in tandem to analyze how the colliding objects behave after the interaction.

Say you have a block with a net external force on it, accelerating toward a block of equal mass which sits at rest. The external force is applied the entire time at the moment of collision and during. One cannot use conservation of momentum or conservation of energy because there's an outside force on the whole system, but I was thinking that during a very short time dt right at the moment of collision you could treat the velocity as constant and the stationary body would obey conservation of momentum over that very short time interval dt, but I can't reason how we represent what happens to the body that is still accelerating and what happens after the interval dt.

I think having been so in the habit of solving these types of problems in a format where the conservation laws hold I am unable to think about how to represent, mathematically, the evolution of the system in a case like this where they do not hold.

Could someone please help elucidate how the equations of motion would be handled in this case?