Q1) A jet-propelled boat, moving with a velocity of 5 m/s, draws water amid-ship. The water is discharged through to jets provided at the back of the ship. The diameter of each jet is 150 mm. The total resistance offered to the motion of the boat is 4905N (500$\times$9.81)

Determine:-

(i) Volume of water drawn by the pump per second,

(ii) Efficiency of jet propulsion

Solution:

Given:

a) Velocity of boat, u = 5m/s

b) Diameter of each jet, d=150mm = 0.15m

c) Area of each jet, $ = \dfrac\pi4(0.15)^2 = 0.01767m^2$

$\therefore$ Total area, $a = 2\times0.01767 = 0.03534m^2$

d) Total resistance to motion = 4905N (500$\times$9.81N)

Propelling force must be equal to total resistance,

$\therefore$ F = 4905N or (500$\times$9.81N)

$F = \rho a(v+u)v$

$\therefore 500\times9.81 = 1000\times0.03534\times (v+5)\times v$

$500 = \dfrac{1000}{9.81}\times 0.03534\times (v+5)\times v$

$500 = 3.6(v+5)v$

$500 = 3.6v^2+3.6\times 5v$

$500 = 3.6v^2+18v$

$0 = 3.6v^2+18v-500$

The above equation is quadratic, so,

$v = \dfrac{-18\pm\sqrt{18^2+4\times 3.6\times 500}}{2\times 3.6}$

$v = \dfrac{-18\pm86.74}{7.2}$

$v = \dfrac{86.74-18}{7.2}$

$v = 34.37m/s$ [-ve value is not possible]

To find:

i) Volume of water drawn by pump per second is equal to the volume of water discharged through the orifices for the back

$ = av_r = a(v+u)$

$ = 0.03534\times (34.37+5.0) = 1.39m^3/s$

ii) Efficiency of jet propulsion

$\eta = \dfrac{2vu}{(v+u)^2}$

$\eta = \dfrac{2\times 34.37\times 5}{(34.37+5)^2}$

$\eta = 0.2217 = 22.17\%$

or

$\eta = 22.17\%$