The first way, I say that in one second, Ω amount of sand falls onto the belt, thus every second 1 2 Ω v 2 is required to continue moving the belt forward (simply by looking at kinetic energy difference) and P = 1 2 Ω v 2. The second way, I saw that F = d ( m v) d t, thus F = Ω v. Finally, P = F v so P = Ω v 2.