differential equation

impluse = change in momentum
ρgxdt=ρ(x+dx)(u+du)-ρxu
gxdt=xdu+udx
gx=x(du/dt)+u(dx/dt)=(xdu/dt)+u^2
d/dx(x^2u^2)=2gx^2
u=0 when x=0
x^2u^2=(2/3)gx^3
u=sqrt(2gx/3)
2u(du/dt)=(2g/3)dx/dt=(2g/3)u
du/dt=g/3
sub t=l,t=(3/g)sqrt(2gl/3)=sqrt(6l/g)
The velocity of the vertical part of the chain when a length x has slipped off the table is sqrt(2gx/3) and that the whole chain has slipped off in time sqrt(6l/g) .