| Fig. 164. |
§ 163. Jet Propeller.—In the case of vessels propelled by a jet of water (fig. 164), driven sternwards from orifices at the side of the vessel, the water, originally at rest outside the vessel, is drawn into the ship and caused to move with the forward velocity V of the ship. Afterwards it is projected sternwards from the jets with a velocity v relatively to the ship, or v − V relatively to the earth. If Ω is the total sectional area of the jets, Ωv is the quantity of water discharged per second. The momentum generated per second in a sternward direction is (G/g) Ωv (v − V), and this is equal to the forward acting reaction P which propels the ship.
The energy carried away by the water
= 1⁄2 (G/g) Ωv (v − V)2.
(1)
The useful work done on the ship
PV = (G/g) Ωv (v − V) V.
(2)
Adding (1) and (2), we get the whole work expended on the water, neglecting friction:—
W = 1⁄2 (G/g) Ωv (v2 − V2).