I posted this same question on my university board and someone kindly posted back the answer below which is hugely informative. Is this completely right? It seems the most detailed description of a collision, conservation of momentum and kinetic energy I've seen.
Your problem should be examined over the course of stages:
1) Why is momentum conserved?
2) What happens in a collision?
3) Is Kinetic Energy always conserved?
4) When it’s not conserved, where does it go?
5) How does a simple equation such as conservation momentum know of heat loss?
1) Newton tells us that a force caused a mass to accelerate (F=ma). This can also be expressed as saying a force causes a mass’s velocity to change over time (F=m(V2-V1)/t). Whenever you see something moving but its motion is not constant or at rest, you know there’s a force involved somewhere. This can also be stated as a force causes a change in momentum over time (F = (mv2-mv1)/t). This can also be expressed as if a force acts on a mass for a duration of time, its momentum will change (F.t = mv2-mv1). Newtons third law shows us that for object 1 to apply a force on object 2, object 2 will apply an equal and opposite force on object 1. If object 1 applies it force on object 2 for a duration t, then object 2 applies the same force on object 1 for the duration t. Therefore each object’s momentum will change by the same amount equally, one’s will increase and the others will decrease. Therefore all collisions cause the same changes in momentum for all objects involved and the total momentum of the system will be constant. Momentum is therefore always conserved due to the symmetry of forces acting on each other by Newton’s third law and because they operate on each other over the same time, they change each other’s momentum by the same amount.
2) One ball of mass M is moving with velocity V towards a stationary ball of equal mass. At the moment of contact, if ball 1’s velocity changed from its velocity V to 0 (dead stop) in zero time, this would require an infinite force. In the same way if the velocity of ball 2 changed from 0 to some other velocity in zero time would take an infinite force too. What is needed is a small amount of time and distance for this process to happen and is provided by the elasticity of the balls involved.
When ball 1 and 2 make contact, there will be an equal and opposite force on each ball. The force of ball 2 on ball 1 will seek to slow it down and compress it so its centre of mass gets closer to that of ball 2’s. The force of ball 1 on ball 2 will seek to speed ball 2 up and give it motion and also compress it. This compression buffer zone allows for ball 2’s stationary velocity to grow upward and ball 1 to decelerate so it’s velocity falls downward. At midway between these two velocities = V/2 both balls are moving at the same speed, ball 1 having slowed down to this from V and ball 2 having gained speed from rest to this. At this point both balls are moving at the same velocity V and they both have potential compression energy stored within them.
The forces involved here are contact forces. This means that when the balls are not touching, the velocity of the balls must either be at rest or at a constant speed as there can be no force on them. This tells us that if ball 1 is ever going to stop dead, it must do so while in contact with a decelerating force from ball 2 during the time they’re in contact. Any speed ball 1 has after ball 2 leaves it, it will continue to have and display uniform motion after the collision.
So ball 1 and 2 are moving at the same velocity V/2 with their centre of masses slightly closer together due to the compression of their structures. They both contain potential energy. Initially the velocity V of ball 1 was all the energy available to the system, during the collision the velocity of the combined mass dropped to half which means the kinetic energy of the system dropped to ¼ of what it was, but as this point when the masses are joined and compressed like this, the mass can be considered to have doubled, and so because KE is dependant linearly on mass, the KE will have dropped by ¼ due to the fall in velocity but doubled due to the doubling of the mass - ¼ x 2 = ½ and so the kinetic energy at this point is now half what it was. So ball 1 had an energy before the collision and during the collision at the point where the balls were both moving at the same speed and were closest together, half of that energy remained in providing motion to the balls and the other half had been converted and stored in the balls themselves to keep them compressed. If half the original Kinetic energy is now stored as PE by both the balls together, then each one must store ¼ of the original KE (half each).
The balls, again moving with V/2 now begin to reform and bounce back against each other. Ball 1 pushes against Ball 2 to accelerate it and Ball 2 pushes against Ball 1 to decelerate it. Ball 1 drops from V/2 to rest, and Ball 2 speeds up from V/2 to V. All the stored PE of the compression between both of them is now converted back to KE and KE is conserved. An easy way to imagine this is of a wall moving at speed V/2 to the right. On the right side of the wall is ball 2 squashed and compressed up against it. On the left side of the wall is ball 1 squashed up and compressed against it. The wall and the balls are moving at the same speed, as ball 2 reforms it pushes away from the wall and moves faster than the wall. Ball 1 recoils leftward and therefore moves slower than the wall it’s just departed from. Ball 1 comes to a dead stop and Ball 2 moves even faster. The symmetry of the forces involved also means momentum is conserved and so Ball 2 speeds away with a velocity V. In every collision, again because of the symmetry of the way forces work, momentum is always conserved. This is always true on a closed system. When Kinetic Energy is also conserved like in the example above, the collision is referred as elastic. In Elastic collisions both momentum and KE are conserved. When KE is not conserved, it just means the energy has been transferred into other forms and if you accounted for all the forms you would find total energy conserved in any closed system, just like momentum.
3) Imagine both balls above again, having smashed into each other and are moving at velocity V/2 each with potential energy stored in their configuration. This time however imagine they are stuck together so that as the balls try to recoil from each other, one pulls in one direction and the other pulls in the opposite direction, the forces cancel and there is no resulting change in the momentum of the combined body at all, it continues to move with V/2.
Where does this energy go then, it was stored PE and it doesn’t look like the balls are able to accept it in order to change their kinetic energy. Imagine a spring on the floor with a ball on top of it. Press the ball downward so the spring underneath it compresses. Let the ball go and the energy stored in the spring throws the ball up into the air giving it kinetic and potential energy. Now imagine the ball is stuck to the spring, as the spring launches the ball upward, it cannot release it so the spring continues to move upward with the ball. Once the string has passed the point of equilibrium, it now begins to stretch and store energy within it as a result of it being pulled apart. The ball will move upward, its motion dampening as all its kinetic energy is being stored in the spring and it will come to rest. The ball will then be sucked downward as the spring releases the energy back to the ball and the scenario will repeat. Essentially the ball will bob up and down, the energy existing half the time in the vibration and half the time stored in either extreme configuration of the stretched of compressed spring. The energy never really goes anywhere it’s just passed back and fourth between KE and PE.
Imagine again the two balls traveling at V/2 stuck together, they both recoil outward just like the spring but they are not free to move and so begin to stretch and store the energy of motion within themselves again. Then they’ll compress together and convert all that KE back to PE. Essentially the balls will be stretching and compressing and all that energy will be bouncing around inside, purely vibrational energy. We know this as heat. The balls will get hot because they cannot separate. How much energy will be lost in this way? Well all the energy they had during the compression was ¼ of the total initial KE, so between them ½ the total KE will be retained within their internal structures.
4) So when two balls smash into each other and stick together, the resulting body will move away at half the speed because the other half of the speed has now been locked inside both of them as heat. Momentum is still conserved but KE is not. However the energy has not been destroyed, it’s still there it’s just not what we call Kinetic energy anymore, it’s called heat energy.
5) How is this process so neatly explained by the conservation of momentum. It’s because this understanding that momentum is always conserved is due to an insight of the symmetry of how forces operate in our universe in action-reaction pairs. All collisions involve forces, all operating in pairs on each other over the same time period, all changing their momentums by equal and opposite amounts. The total momentum therefore is constant. When you feed a problem into this equation such as an inelastic collision of one ball moving with a speed colliding and sticking to another ball so that they both move with half the speed, all the conservation of momentum equation sees is that there was one ball moving, now there’s two balls moving at half the speed. There must have been forces involved, forces act in pairs and therefore momentum must have been conserved. The law doesn’t tell you what happened to the energy lost, it cannot speak to that, it knows about momentum only. It basically just tells you that it knows that our universe consists of forces that never waiver in their behaviour of acting in equal opposite pairs, it calculates the outcome of a collision based on that known behaviour. It’s like asking a man if he saw a car crash and he tells you he only saw two cars approaching each other at an intersection. That’s all the information you’re going to get from him, yet you see a wreck of two cars at the intersection and you know there’s been a crash. You put the pieces together yourself, the man can only speak to what he knows and the law of conservation of momentum can only speak to its understanding of conservation of momentum. In a collision you must analyse the result of what happened to this energy based on your knowledge of heat mechanics. If it has to, nature will convert energy into different forms to maintain this symmetry and the sanctity of Newtons laws.
