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.