Jet propulsion of a ship: propelling force with side inlets at right angles to motion Given: water is drawn through inlet orifices at right angles to the ship's direction of motion. Symbols: a = jet area, V = jet velocity relative to still water (issuing from the ship), v = ship speed, Vr = relative velocity between the jet stream and the ship (= V + v as stated). Find the correct expression for the propelling (thrust) force based on momentum. Choose the correct option.

Concept / Approach
Thrust in jet propulsion comes from the rate of change of momentum in the direction of motion. Mass flow rate through the ship's system (as seen on the ship) = ρ a Vr. Velocity component change along the ship's axis between inlet and exit = (V − v) − 0, because the inflow through side inlets has no axial component.

Step-by-step
Thrust F = (mass flow rate) × (axial velocity change) F = (ρ a Vr) × (V − v)

Verification
If the ship is stationary (v = 0), then Vr = V, and F reduces to ρ a V × V = ρ a V², as expected for a stationary nozzle.

Common pitfalls
Adding velocities instead of taking the difference (V − v) in the momentum change term. Using ρ a V instead of ρ a Vr for the mass flow seen by the moving ship.

Final Answer
ρ a Vr (V − v)