“Huygens problem”. Consider the ball with mass M moving with velocity V towards another ball with mass m that stays at rest. After the central collision, the second ball acquires the velocity
\[ v = { \frac { 2 M } { m + M } } V .\tag{11} \]
This expression can be obtained using the momentum and energy conservation laws for the two-body system. One can observe that \(V \leqslant v \leqslant 2 V\) as far as \(0 \leqslant m \leqslant M\) One may ask under what conditions the limit \(v \leqslant 2 V\) can be broken to make v arbitrary large. A possible solution is to insert a chain of balls staying at rest with intermediate masses \(m _ { 1 } , . . . , m _ { n }\) such that \(m < m _ { 1 } < . . . < m _ { n } < M\) between the two original bodies, and to transfer the kinetic energy of the moving ball to the ball with mass m through a sequence of intermediate central collisions.
Applying Eq.(11) to the sequence of central collisions between the balls, deduce how one should choose the masses \(m _ { 1 } , . . . , m _ { n }\) to yield the maximal velocity of the ball with mass m.
Assuming \(m \ll M\) , investigate the limit \(n \to \infty\)