In regard to the Maxwell speed distribution, you might wonder why all the molecu

Question

In regard to the Maxwell speed distribution, you might wonder why all the molecules in a gas in thermal equilibrium don’t have exactly the same speed. After all, when two molecules collide, doesn’t the faster one always lose energy and the slower one always gain energy? And if so, wouldn’t repeated collisions eventually bring all the molecules to some common speed? Describe an example of an elastic billiard ball collision in which this is not the case: the faster ball gains energy and the slower ball loses energy. Include numbers and be sure that your collision conserves both energy and momentum.

Explanation / Answer

consider two billiard balls of equal masses m, one moving with velocity v(cos(theta)i + sin(theta)j) and the other moving with velocity u(cos(phi)i + sin(phi)j)
also assume v > u
so initial momentum of the system
m(v(cos(theta)i + sin(theta)j + cos(phi)i + sin(phi)j) = pi

now when these balls collide
let final velocities of the balls be
v'(cos(theta')i + sin(theta')j) and u'(cos(phi')i + sin(phi')j)
hence final momentum
m(cos(theta')i + sin(theta')j + cos(phi')i + sin(phi')j) = pf

from conservation of momentum
(v(cos(theta)i + sin(theta)j) + u(cos(phi)i + sin(phi)j) = v'(cos(theta')i + sin(theta')j) + u'(cos(phi')i + sin(phi')j)

also, for perfectly elastic colision
coefficient of restitution = 1
then
[v'(cos(theta')i + sin(theta')j) - u'(cos(phi')i + sin(phi')j)] = u(cos(phi)i + sin(phi)j) - v(cos(theta)i + sin(theta)j)

hence we get the following equations
vcos(theta) + ucos(phI) = v'cos(theta') + u'cos(phi')
vsin(theta) + usin(phI) = v'sin(theta') + u'sin(phi')
v'cos(theta') - u'cos(phi') = ucos(phi) - vcos(theta)
v'sin(theta') - u'sin(phi') = usin(phi) - vsin(theta)

2v'cos(theta') = 2ucos(phi)
2v'sin(theta') = 2usin(phi)

so we get v' = u
hence the velocities are exchanged for billiard balls with equal mass
hence for equal masses one cannot say that the fastger ball will gain energy and the slower ball will not
but for unequal masses the faster ball can gain speed and the slower one can lose speed dependiong on the mass

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