Why do colliding objects have equal changing velocities if momentum is conserved?

[Werg22]

The theorem of conservation of momentum states that the quantity of momentum is always the same. When two objects collide, during the interval of time they touch each other their velocities are the same but are in the same time changing, if they are both changing but remaining equivalent, why is it in contradiction with the theorem?

 

[Tide]

The contradiction is your assumption that their velocities are the same during the interval the objects touch each other. Real objects are deformable so during contact one may be speeding up as the other slows down. Moreover, they certainly do not have the same "tangential" velocity during that interval. In the limit of idealized perfectly rigid bodies the time of contact goes to zero and velocities change by impulse.