Energy of various systems in an inelastic collision

 

I was asked about the situation above, in which two carts of different masses are released from rest and roll down frictionless ramps.  The carts collide and stick together on the flat surface.

Let's treat this as a goal-less problem:

Edna, Bertha, and Anthony by @Aldescary

Which way do the carts move after they collide?  

Well, Anthony (whom Edna calls a mean hippopotamus) says that the carts don't move after collision - they both have the same energy but in opposite directions, which cancel.

Oy.  So many things wrong with Anthony's answer.  Let's start with the fact that energy is a scalar quantity - energy can't have direction at all, let alone "cancel" other forms of energy.  And whenever we see a collision, energy should never be the first port of call - momentum should be.

It is true that the gravitational energy of each earth-cart system is converted to kinetic energy at the bottom.  And since the gravitational energy 2mgd is the same for both earth-cart systems, each cart will have the same kinetic energy before collision.

However, the momentums of each cart will be different.  I like to use the shortcut equation K = p^2 / 2m in this case to see that with the same kinetic energy, the cart with a greater mass will also have greater momentum.  You could also convert gravitational energy to kinetic energy to show that the speed at the bottom will root 2gh for both carts with mass canceling; then by p = mv twice as much mass with only 1.4 times as much speed means bigger momentum.

So the cart moving left has larger momentum than the cart moving right, meaning the two-cart system has a leftward momentum.  System momentum must be the same after collision as before, so the momentum is still leftward after collision... and that's the way the stuck-together carts will move.

Is mechanical energy of the both-carts-and-earth system conserved from release to just BEFORE the collision?

Anthony says mechanical energy is potential plus kinetic energy, and is always conserved, so yes.

Well, even a blind squirrel, or hippopotamus, finds a nut once in a while.  Anthony is pretty much correct.  Mechanical energy is conserved when no work is done by external forces and when no internal energy conversion occurs.  Here, the only external force acting on the carts-earth system is the normal forces of the surfaces on the carts.  These forces are perpendicular to the carts' motion, and so do no work.  Mechanical energy is, in fact, conserved here!

Is mechanical energy of the both-carts-and-earth system conserved from release to just AFTER the collision?

Anthony is perturbed... he already answered this question!  Mechanical energy is conserved, period, full stop, end of sentence.  Hemph.

Oh, Anthony... when carts collide and stick together, they undergo an inelastic collision by definition.  Mechanical energy may never be conserved in an inelastic collision - rather, some mechanical energy must be converted to internal energy.  

Is mechanical energy of the both-carts-only system conserved from release to just AFTER the collision?

Anthony says he's done answering these tricky questions involving systems.  He's gonna put his head down on his desk and listen to Edna for a change.

Edna thinks this one is pretty simple... because an object by itself can only possess kinetic energy!  (Potential energy can only exist when a spring or the earth is incl
uded in the relevant system.)  So the mechanical energy of the carts is just their kinetic energy.  On release from rest, the carts have no speed and therefore no kinetic energy.  After collision, the carts are moving, so they have kinetic energy.  The KE has gone from zero to not-zero, and therefore has increased.