Consider an object with mass energy E1 approaching at speed and closing on an object of mass energy E2 regarded as being stationary. The energy desires of the mass energy in the approaching structure are such that they do not welcome the close approach and they release as photons some of their mass energy E1 in an appropriate directional way to slow and reverse the approach. The approached object does not want these high volume photon energies not does it welcome the close approach and it too is releasing what would otherwise be a growing mass energy E2 to also oppose, slow and reverse the approach.
The above is my description of the energy interactions that slow and reverse structures in a collision. The interactive thrusts they provide are not steady. They get higher as the objects get nearer and reduce as the objects move apart. However at any point in time within the duration of the collision they are equal and opposite.
Those ever changing but common thrusts act on the mass energies E1 and E2 so as to slow them. If E1 is small relative to E2 they will slow it more rapidly. The motion changes of the energies are proportional to the common thrust as distributed over and acting per unit of mass energy. Since mass and energy are just different units of the same thing it means that in any one direction the mass times the velocity change of one object is equal to minus the mass times the velocity change of the other object. This wordy description explains in energy terms why the conservation of momentum law, derived mathematically above works.
Many are fascinated by Newton’s cradle and so I will try to explain in energy terms why it does what it does. It comprises a number of balls on wires in close contact. Most will know that if we draw back one ball and release it the ball at the other end of the cradle will respond with a near identical motion but in the opposite direction. If we raise and let two or three balls go the response similarly involves two or three balls.
When we lift and release ball 1 the desire of its particles for earth energies accelerate it earthward, albeit restrained by its support wire. It makes a curved approach to ball 2 particles at a gathering speed. A collision as described above now occurs between balls 1 and 2. The energy linked particles of ball 1 are rapidly slowed whilst the photon energy linked particles in balls 2, 3, 4 and 5 are put in motion.
In a perfectly elastic collision (no such thing) the internal photon energies acting within the balls hold their particles rigidly apart and the motion of any one particle immediately affects the motion of others. In the case of our steel balls the internal photon energies act at light speed between particles setting them rapidly in motion with very little compression of the distances between them. There is little mass energy added to the balls as a result of the collision. The photon energy exchanges also pass between the particles of the close contact balls. The energy linked particles of ball 5 are not restrained like those of balls 2 to 4 and so its motion is almost a reflection of the motion of ball 1.
If the balls had been made of plasticine some collision energy would come to reside in the compressed particle structure as added mass energy. The linear momentum along the line of the balls is still conserved in this inelastic collision because the linear thrust from particle to particle and ball to ball remains fixed.
Be aware that during the time of a collision neither mass nor velocity are fixed but their product in a specific direction is because the directional thrust is. If we want to measure mass and directional velocity we should do so when they are most stable immediately before engaging in the collision or immediately after release from the collision.
You may ask why just ball 5 moves when ball 1 is released yet balls 4 and 5 will move when 1 and 2 are released. I have explained above how all particles and balls are set in motion in response to the collision energy pulses. When two balls approach the cradle ball 2 starts to interact with ball 3 producing a pulse that passes along the cradle and releases ball 5. However at photon light speed the slowing ball 2 is interacting with ball 1 and so adding to the pulse. The pulse is not an instantaneous one but one that rises and falls in a short time and probably with two peaks because balls 1 and 2 are separate energy linked structures. The pulse continues after ball 5 has been set in motion and sets in motion the now free of restraint ball 4.
The balls are photon energy linked to one another but not bound together like their particle contents. Observe how after a few Newton’s cradle swings the motions become less perfect as the balls lose a little of their close contacts with one another.
Colliding energies make every effort to stay apart because they value the energy efficiency of their independent structures. The interactions may cause them to take on board added thermal energies but they don’t and they would not want to take on board kinetic energies of motion.