Yes, I do remember. Of course, it's true, but it also depends on how you look at it. If you include the earth in the system, then momentum is conserved, but that's not saying much of use. If a fast car hits a solid wall and comes to a dead stop, there's nothing to be gained by claiming momentum is conserved in the car-earth system. OK, momentum is not conserved, because of how we define our system.
If a car hits a second (stopped) car and the two, fused together, hit a wall, can we say momentum is conserved? In the first collision, yes, in the second collision and overall, no. What if the cars are in the process of fusing (i.e. the first collision is not over) when the second collision begins, as would be the case if the stopped car were parked an inch from the wall? Momentum is only conserved up until the imposition of an external constraint force.
But wait... the frictional force of the tires against the road is an external force. Put the parked car a mile away from the wall and the pair of fused cars will never reach it. Momentum was not conserved in the first place, after all. Now, what about having the parked car butted up against the wall, but have the wall just be a flimsy fence that breaks away? The head swims.
In this model, I say yes, momentum is conserved in slab collisions - more or less - and that's correct dynamics for this configuration. Only 'more or less' because there are connections that consume energy when they break, and that just happens to be in the same small region of time and space as the collision. These connections are then analogous to the breakaway wall in the example above. If the slabs were just suspended in space at equal distances, zero g and without connections, no question the collisions would be between individual slabs, not 'blocks', and momentum is conserved.
Add weak connections; not enough to hold the slabs against gravity. Momentum not conserved, technically, but in the limit of very weak connections, yes. Make them stronger, until strong enough to arrest immediately in gravity, no conservation at all.
If one takes the collision between slabs as instantaneous, then no distance is traveled during the collision. The bodies emerge from the collision together at the new, reduced velocity. Because there is no displacement, forces of constraint can do no work during the collision; connections are broken after subsequent displacement. With respect to individual bodies, momentum is conserved.
In actuality, this simulator uses a thin skin around bodies to allow some interpenetration and finite collision distances, as such is more realistic than a purely analytical computation based on an idealized assumption. Also, the connections break over a small but measurable distance at the time of collision so, practically speaking... momentum is not conserved.
Hahaha, isn't this grand?
This is what matters: while a connection survives, it transmits force to the next solid member, and so on. Once broken, in this discrete model, transmitted/resistive force due to connections goes to zero. Slab inertia remains
An upper block may have X times the KE necessary to fail the first connection below, but it also has to accelerate the topmost lower slab to the upper block speed very quickly or the differential displacement will exceed the connection limit for the bottom of Zone C. We know there's enough energy to do so. The inelastic collision between the interface slabs only brings the fused pair of slabs to half the Zone C speed. Zone C can only be slowed by upward force applied through the surviving lowest connection
, it is a mistake to take the collision as happening between all the slabs of Zone C and the one slab below. This same lowest connection applies downward force from Zone C motion to the pair of slabs in contact, and their inertia provides a reaction force.
If the newly formed Zone B, now consisting of one slab, and Zone C do not reach a common speed right away - by virtue of force transmitted through the single lowest connection of Zone C
- that connection fails and Zone B then has two members. Zone C is still moving faster than Zone B. Crushing up.
How can the first slab of Zone B (fomerly top of Zone A) receive any more impulse from Zone C than the connection strength can provide before failure? Well, it can't. So it really doesn't matter if the upper block has a million times the KE to fail one connection, the limit to the force it can apply is the connection strength. Hence, simulations with varying connection strength (stronger at the bottom) will fail the lowest connection of Zone C under most circumstances, and crush up whether or not they continue to crush down.
This model, conceptually simple with its 'incompressible' inelastic slabs separated by empty space, has characteristics far from everyday experience with materials and structure. The simplicity allows exploration of collapse dynamics without too many factors muddying the waters, but is correspondingly limited in its output and application. It is arguably closer to reality than a continuous, uniform
mass distribution, though, so its lessons are worth considering. This is as close to the blocks being rigid bodies as one can get without DEFINING them to be rigid, and they're just not rigid. Only the slabs are.
Major_Tom wrote:The thread is great.
Thanks. Excellent question.