1. A ship is driven through water, by the application of jet propulsion principle.

2. A propulsive force is exerted on the ship when the jet of water is discharged at the back or from the back of the ship (which is also called as stern).

3. The ship carries centrifugal pumps which helps how to draw the water from the surrounding sea.

4. This water which is drawn is discharged through the orifice which is provided at the back of the ship in the form of jet.

5. This helps to prepare the ship in opposite direction of jet.

6. The water from the surrounding sea by the centrifugal pump is taken out in two ways as follows:

• Through inlet orifice which are at right angles to the direction of the motion of the ship.

• Through the inlet orifices, which are facing the direction of motion of the ship.

1st case:-

Jet propulsion of the ship when the inlet orifice are at right angles to the direction of the motion of the ship.

Figure shows a ship which is having the inlet orifice at right angles to its direction.

Let,

v = absolute velocity of jet of water coming at the back of the ship.

u = Velocity of ship

$v_r = (v+u)$

As the velocity 'v' and 'u' are in the opposite direction, hence relative velocity will be equal to the sum of these two velocities.

Mass of water issuing from the orifice at the back of the ship = $\rho av_r = \rho a(v+u)$

where

a = area of jet of water.

$\therefore$ propulsive force exerted on the ship

$F = \rho av_r[v_r-u]$

$F = \rho av_r[(v+u)-u]$

$F = \rho a v_r\times v..................(1)$

Work done per second

$= F\times u$

$= \rho av_r\times v\times u.................(2)$

Now, if the loss of head due to friction in the intake and ejecting pipes is considered then loss of energy due to friction will be $(w\times hf)\text{ or }[(m\times g)\times hf]$

where,

$hf =$ loss of head due to friction in pipes.

$W =$ Weight of water coming out from jets

$W = m\times g$

mass, $m = \rho\times a\times v_r$

The pump in this case has to work more to supply extra power to overcome the pipe losses. Hence the output of the pump should give the jets a relative velocity and also overcome the velocity loss in pipes.

$\therefore$ Output of pump = kinetic energy of jet + power required to overcome pipe losses

$= \dfrac12Mv_r^2+(m\times g)\times hf$

Mass, $m = \rho\times a\times v_r$

$\therefore \text{output} = \dfrac12m{v_r}^2 + (m \times g)\times hf$

$\therefore \text{output} = \dfrac12(\rho av_r)v_r^2+(\rho.a.v_r\times g)\times hf$

Efficiency of propulsion will be,

$\eta=\dfrac{\text{work done by jets on ship}}{\text{Output of pump}}$

$\eta=\dfrac{F\times u}{\dfrac12(\rho av_r)v_r^2+(\rho.a.v_r\times g)\times hf}$

$\eta=\dfrac{(\rho.a.v_r\times v)\times u}{\dfrac12(\rho av_r)v_r^2+(\rho.a.v_r\times g)\times hf}$

$\eta=\dfrac{v\times u}{\dfrac 12.v_r^2+ghf}$

$\eta=\dfrac{2\times v\times u}{v_r^2+2ghf}$