It's just Bernoulli's principle which is nothing more than $F=ma$ in a fluid.
Force in a fluid is called pressure.
The only way a fluid can accelerate is by going from high pressure to low (i.e. feeling a force pushing it forward).
Similarly, the only way it can slow down is by going from low pressure to high (feeling a force pushing it backward).
Do the whole experiment horizontally, so you can ignore potential energy.
Energy and momentum are linked by velocity.
So $F=ma$ is both conservation of momentum and conservation of kinetic energy.
Excuse me while I do a little math to show that $F=ma$ conserves energy:
An increment of work is $Fdx$.
An increment of distance is $dx = vdt$.
$F=ma$ says $F=mdv/dt$.
So the increment of work is $Fdx = ma \times dx = mdv/dt \times vdt = mvdv$.
The increment in kinetic energy is $dKE = 1/2 m(v^2 + 2vdv + (dv)^2 - v^2) = mvdv$,
(because you can ignore $(dv)^2$).
So $Fdx = dKE$.