Δ Δ E pot,gravity = Δ Δ m g z 2 − − Δ Δ m g z 1 . Now, the work by the force of gravity is opposite to the change in potential energy, W gravity = − ΔE pot,gravity : while the force of gravity is in the negative z -direction, the work—gravity force times change in elevation—will be negative for a positive elevation change Δ z = z 2 − z 1, while the corresponding potential energy change is positive. So: W gravity = − − Δ Δ E pot,gravity = Δ Δ m g z 1 − − Δ Δ m g z 2 . And therefore the total work done in this time interval Δ t is W = W pressure + W gravity . The increase in kinetic energy is Δ Δ E kin = 1 2 Δ Δ m v 2 2 − − 1 2 Δ Δ m v 1 2 . Putting these together, the work-kinetic energy theorem W = Δ E kin gives: Δ Δ m p 1 ρ ρ − − Δ Δ m p 2 ρ ρ + Δ Δ m g z 1 − − Δ Δ m g z 2 = 1 2 Δ Δ m v 2 2 − − 1 2 Δ Δ m v 1 2 or 1 2 Δ Δ m v 1 2 + Δ Δ m g z 1 + Δ Δ m p 1 ρ ρ = 1 2 Δ Δ m v 2 2 + Δ Δ m g z 2 + Δ Δ m p 2 ρ ρ . After dividing by the mass Δ m = ρA 1 v 1 Δ t = ρA 2 v 2 Δ t the result is: 1 2 v 1 2 + g z 1 + p 1 ρ ρ = 1 2 v 2 2 + g z 2 + p 2 ρ ρ or, as stated in the first paragraph: