Two particles \(A\) and \(B\), of masses \(m\) and \(km\) respectively, lie at rest on a smooth horizontal surface. The coefficient of restitution between the particles is \(e\), where \(0 < e < 1\). Particle \(A\) is then projected directly towards particle \(B\) with speed \(u\).
Let \(v_1\) and \(v_2\) be the velocities of particles \(A\) and \(B\), respectively, after the collision, in the direction of the initial velocity of \(A\).
Show that \(v_1 = \alpha u\) and \(v_2 = \beta u\), where \(\alpha = \dfrac{1 - ke}{k+1}\) and \(\beta = \dfrac{1+e}{k+1}\).
Particle \(B\) strikes a vertical wall which is perpendicular to its direction of motion and a distance \(D\) from the point of collision with \(A\), and rebounds. The coefficient of restitution between particle \(B\) and the wall is also \(e\).
Show that, if \(A\) and \(B\) collide for a second time at a point \(\frac{1}{2}D\) from the wall, then
\[ k = \frac{1+e-e^2}{e(2e+1)}\,. \]
Three particles \(A\), \(B\) and \(C\), of masses \(m\), \(km\) and \(k^2m\) respectively, lie at rest on a smooth horizontal surface in a straight line, with \(B\) between \(A\) and \(C\). A vertical wall is perpendicular to this line and lies on the side of \(C\) away from \(A\) and \(B\). The distance between \(B\) and \(C\) is equal to \(d\) and the distance between \(C\) and the wall is equal to \(3d\). The coefficient of restitution between each pair of particles, and between particle \(C\) and the wall, is \(e\), where \(0 < e < 1\). Particle \(A\) is then projected directly towards particle \(B\) with speed \(u\).
Show that, if all three particles collide simultaneously at a point \(\frac{3}{2}d\) from the wall, then \(e = \frac{1}{2}\).