N particles P1, P2, P3, ..., PN with masses m, qm, q²m, ..., q-¹m, respectively, are at rest at distinct points along a straight line in gravity-free space. The particle P₁ is set in motion towards P2 with velocity V and in every subsequent impact the coefficient of restitution is e, where 0 <e < 1. (i) Show that after the first impact the velocities of P₁ and P2 are respectively. (i-eg)v 1+q eq V and (++) v. V, (ii) Show that, if q ≤ e, then there are exactly N − 1 impacts. (iii) Show further that, if q = e, then the total loss of kinetic energy after all impacts have occurred is me (1-e-1)V2. 1999 Paper II