The water in a jet propelled boat is drawn through inlet openings facing the direction of motion of the ship. The boat is moving in sea-water with a speed of 40 km/hr. The absolute velocity of the jet of the water discharged at the back is 40 m/s and the area of the jet of water is 0.04 m². Find the propelling force and efficiency of propulsion.

Jet Propelled Boat Analysis

Problem Statement

Given Data & Constants

Speed of boat, $u = 40 \, \text{km/hr} = \frac{40 \times 1000}{3600} \approx 11.11 \, \text{m/s}$
Absolute velocity of jet, $V = 40 \, \text{m/s}$
Area of jet, $A = 0.04 \, \text{m}^2$
Density of sea-water, $\rho \approx 1025 \, \text{kg/m}^3$

Solution

1. Calculate Mass Flow Rate and Relative Velocity

The mass of water discharged per second is based on the absolute velocity of the jet.

$$ \text{Mass flow rate, } \dot{m} = \rho A V = 1025 \, \text{kg/m}^3 \times 0.04 \, \text{m}^2 \times 40 \, \text{m/s} = 1640 \, \text{kg/s} $$

The change in velocity that produces the thrust is the relative velocity of the jet with respect to the boat.

$$ \text{Relative Velocity, } V_r = V - u = 40 - 11.11 = 28.89 \, \text{m/s} $$

Propelling Force of the Boat

The propelling force is the rate of change of momentum, which is the mass flow rate multiplied by the relative velocity.

$$ \text{Force, } F = \dot{m} \times V_r $$ $$ F = 1640 \, \text{kg/s} \times 28.89 \, \text{m/s} \approx 47379.6 \, \text{N} $$

Efficiency of Propulsion

The standard formula for the efficiency of a jet-propelled boat where water is taken from the surrounding medium is:

$$ \eta = \frac{2u}{V+u} $$ $$ \eta = \frac{2 \times 11.11}{40 + 11.11} = \frac{22.22}{51.11} \approx 0.4347 $$

Final Results:

Propelling force of the boat: $ \approx 47.38 \, \text{kN} $

Efficiency of jet propulsion: $ \approx 43.5\% $

Explanation of Jet Propulsion

Propelling Force: The force that moves the boat forward is a reaction force based on Newton's Third Law. The pump on the boat takes in water (which is already moving relative to the world at the boat's speed) and accelerates it, shooting it out the back at a higher absolute speed. The force required to cause this change in momentum for the water creates an equal and opposite force on the boat, pushing it forward.
Efficiency: The propulsive efficiency measures how well the energy added to the water is converted into useful work to move the boat. The formula $\eta = 2u / (V+u)$ shows that efficiency is maximized when the boat's speed ($u$) is as close as possible to the jet's speed ($V$).