# Approaches to Sand on a Conveyor Belt

Say we are given a conveyor belt with sand falling onto it at rate $\Omega$. I am trying to find the power it takes for the belt to operate if it goes forward with constant velocity $v$, but using two different approaches I get two different answers.

The first way, I say that in one second, $\Omega$ amount of sand falls onto the belt, thus every second $\frac 1 2 \Omega v^2$ is required to continue moving the belt forward (simply by looking at kinetic energy difference) and $P=\frac 1 2 \Omega v^2$

The second way, I saw that $F=\frac {d(mv)} {dt}$, thus $F=\Omega v$. Finally, $P=Fv$ so $P=\Omega v^2$.

So which one is it: $P=\Omega v^2$ or $P=\frac {1} {2} \Omega v^2$, and where did I make an incorrect assumption in my two methods?