I’ve given the falling brick a bit of thought, but not as much as I would like …… (the weather was good enough for me to do a couple of training flights yesterday and a test today on a new pilot who is just finishing his training on 737s with Ryan Air – he passed !).
Anyway, back to the falling brick. These are my current thoughts.
There are lots of unknowns; and collisions in the real world are never perfectly elastic nor inelastic. Take the lift material for example. If the “brick” and “Lift” were both made of solid, hard steel, the collision could be close to elastic, with virtually no loss of kinetic energy. The lift is being acted upon by an external force, ie the hoist cable. This, together with the mass of a typical lift, would mean the lift continues at a steady speed regardless of the impact of the brick. There are other considerations but these will serve enough to camouflage my uncertainty about tackling this problem.
I have assumed that the lifts move at constant, steady speed throughout, with no loss of kinetic energy and no change in momentum. I have assumed that the brick comes to rest relative to each lift, then continues to move with the lifts. In doing so, that is where the conservation of momentum occurs and where, to all intent and purpose the entire kinetic energy of the brick is dissipated (as heat and sound when distorting the lift cage).
The loss of kinetic energy is proportional to the (relative) speed at impact, (actually the speed squared). Now this (relative speed) is where I might be wrong ( ok , I might also be wrong elsewhere !!). The relative speeds at impact are the same for the down-going and up-going lifts
Sometimes it’s worth trying a couple of numerical examples before launching into algebraic equations. So I considered a brick 1kg in freefall at 10m/s/s crashing into (a) an up-going lift 1,000kg travelling at a steady 2m/s and a down-going lift travelling at 2m/s.
First I did it assuming the lifts passed each other 45m below the released brick, then at 80m. These coincide with 3 secs and 4 secs of freefall at 10m/s/s, with the unrestricted brick travelling at 30m/s and 40m/s respectively. In each case, the collision with the up-going brick will occur before that height and time, with the brick travelling more slowly and with the down-going lift after that height and time with the brick travelling slightly faster…
In the 45m case the impact speeds of the brick are 28.07m/s and 32.06m/s whilst in the 80m case they are 38.05m/s and 42.05m/s.
Now clearly, the bricks that hit the descending lift are travelling somewhat faster than those that hit the up-going lifts. So the kinetic energy of the bricks at the down-going lift will always be greater than at the up-going lifts. The usual equations and graphs illustrate and support this.
I’ll give it a bit more thought, but relative speeds seem to be a more realistic option rather than brick-only speeds. At the moment.