[Lukasz.Bartosik]

Hi everyone
I've got a few problems that need rather urgent solving I would appreciate if anyone could post anwsers to one or more of the following problems:

Ex.1 A ball falls on the flour from the height h. After each bounce the velocity is decreased by a factor
a<1. How long the ball bounces before it stops completely? Earth acceleration is denoted by g.

Ex.2 Consider a system in equilibrium with a thermostat at temperature T. The energy of the system
can assume two values, -E or E. The degeneracy of the states with the energies -E and E is N-fold and
2N-fold respectively.
Calculate the average energy (a) for T=0K (b) for T∞

Ex.3 Consider a crystalline lattice consisting of N sites. N1=N1−x sites are occupied randomly by
noninteracting atoms, where 0<x<<1. Calculate the entropy of the system for N∞ .
(Stirling formula: for N∞ ln N !≈N ln N−N )

Thanks a lot in advance

[Trave11er]

Hello Lukasz.

Ex.1 It seems the ball will bounce forever, because velocity never reaches zero. After n bounes its velocity will be v*a^n, but whatever is n, the ball will never stop COMPLETELY

[Borek]

I would appreciate if anyone could post anwsers to one or more of the following problems

Please read forum rules.

[Borek]

It seems the ball will bounce forever, because velocity never reaches zero. After n bounes its velocity will be v*a^n, but whatever is n, the ball will never stop COMPLETELY

No. Number of bounces will be infinite, but they will happen in a finite time.

[Trave11er]

Borek,
Can you explain why do you think so, because to me it seems that atoms will bounce forever. Surely, the height grows smaller and smaller until it becomes infinitely small, but as the speed is never zero, the height will never reach zero either, thus bouncing will never end.

[Lukasz.Bartosik]

I would appreciate if anyone could post anwsers to one or more of the following problems

Please read forum rules.

Sorry for breaking the rule about the topic's name, as for the other rules this is NOT homework of any kind. Also I have attempted to solve the problems and my conclusions ( as far as I remember) are: the ball will jump infinitely long, ex2. Only the lowest energy states will be occupied at 0K and all of the states will be equally occupied in T->infinity ( i don't know where to go from there though)
ex.3 I have no idea where to start.

Once again I'm sorry for breaking the rules but I'm in a bit of chaos now.

[Borek]

Calculate time of n^th bounce t_n, assuming first bounce time was some t₀, then sum times of bounces from n=0 to infinity.

[Trave11er]

This is a solution to the first problem

[Trave11er]

oops...I meant the speed of ball after the first bounce not the second at the end of the page