to the billiard inside an acute triangle with the angles
, / lm1 +ra
2
+ ra3\ .
1 0 Q
arctan m a , i = 1, 2,
V V raim2ra3 /
Remark 1.7. Exercise 1.4 and Example 1.6 provide m
tems, isomorphic to the billiards inside a right or an
It would be interesting to find a similar interpretation
triangle.
Exercise 1.8. This problem was communicated by S.
pose 100 identical elastic point-masses are located so
one-meter interval and each has a certain speed, not le
either to the left or the right. When a point reaches
the interval, it falls off and disappears. What is the lo
waiting time until all points are gone?
In dimensions higher than 1, it does not make sen
point-masses: with probability 1, they will never collid
considers the system of hard balls in a vessel; the bal
the walls and with each other elastically. Such a syst
interest in statistical mechanics: it serves a model of i
In the next example, we will consider one partic
this type. Let us first describe collision between tw
Let two balls have masses mi,ra2 and velocities V\,V
specify the dimension of the ambient space). Conside
of collision. The velocities are decomposed into the
tangential components:
vi = vri+v\, 2 = 1,2,
the former having the direction of the axis connecting
the balls, and the latter perpendicular to this axis. In
tangential components remain the same, and the radi
change as if the balls were colliding point-masses in th
as in (1.1).
Exercise 1.9. Consider a non-central collision of two i
balls. Prove that if one ball was at rest, then after th
balls will move in orthogonal directions.
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