Justin clarified:
"The question had only to do with sand falling on a stationary scale."
Then simply consider that the mass dm changes momentum from v*dm (downward),
to zero, in a time dt. This requires an upward force from the scale of
v^dm/dt.
It's that simple - the recipe "dp/dt = mdv/dt + vdm/dt" is not really
helpful here. We are not here dealing with the dp of a variable mass body,
but with the dp of the single mass dm. We are using the FIRST term, with m
=> dm. It is an impulsive interaction and is best modeled as Impulse
=momentum change:
Fdt = vdm (Be clear, dm is not here the change in a mass - it IS the mass
under consideration.)