Energy and momentum conservation

[aaaa202]

Consider a point mass of mass m going with velocity v towards a point mass also of mass lying still.

Now conservation of momentum allows any combinations og mass times velocities that add to the total momentum before the collision. So for instance ½mv + ½mv would be good. This is where the masses collide until they have the same velocity. For me it makes sense that this would happen since the first mass is not able to push against the other as soon as the mass originally lying still is going faster than the other - which happens infinitesimally later than when their velocities are the same.

However, if energy was to be conserved the above would not hold, since E is proportional to v squared. My question is: How can energy have influence on when our two masses geometrically are able to push against each other or not? - like if energy was conserved in the above they would still be pushing against each other even though the first particle would be slowing down as seen from the second particles frame.

 

[tiny-tim]

hi aaaa202!

… How can energy have influence on when our two masses geometrically are able to push against each other or not? - like if energy was conserved in the above they would still be pushing against each other even though the first particle would be slowing down as seen from the second particles frame.

if energy was conserved, the first mass would stop dead

even if the first mass is much heavier, there is still no question of it pushing the second mass if energy is conserved

change to a frame of reference in which the centre of mass is stationary …

if both momentum and energy are conserved, then obviously the velocities simply change places, ie a "perfect bounce"! ​

 

[aaaa202]

okay but let's slow time down and look at what happens. Half time during the collision the two masses both move in the same direction with same velocity. a split second after that the mass originally lying still will have greater velocity than the first. Conclusion: The masses are unable to touch either and thus interact with forces. But if energy is conserved they still do that - how do you explain that?

 

[Philip Wood]

(1) You are asking about elastic collisions in which kinetic energy is conserved. For 'macroscopic' bodies (i.e. those made up from loads of atoms) collisions are never perfectly elastic. Some of the initial kinetic energy is transferred to increased random vibration of the atoms. But energy is conserved.

(2) I think you've set out a most interesting paradox: how can one body, A, transfer all its KE to another body, B (initially stationary) if, half-way through the process, B starts to separate from A when it's got only half of A's velocity. Hope I've grasped it.

The paradox arises, I think, from your two particle model. Either the particles are touching and exerting forces on each other, or they're not touching, and they go from touching to not touching in an infinitesimal distance. This isn't realistic. In practice the two bodies will have finite size and will deform. They will repel each other (to a greater or lesser extent) throughout the time they're deformed, including after the instant when their centres of mass had the same velocity.

With two (spinless) positively charged particles 'colliding it's even easier to see what's going on; the collision process is one of repulsion over a period of time.