Question

The wind over the ocean blows straight North with a speed of 10.0 m/s. A sailboat sails straight North with a steady speed of 6.00 m/s, and its sail is perpendicular to the direction of the wind. Suppose the boat deploys a second sail of the same area and drag coefficient perpendicular to the wind above the first sail. What is the new steady speed ? of the boat? Assume quadratic drag for both wind blowing into sails and body of the boat moving through the water. Neglect size of the boat compared to the size of sails when considering air drag.

please provided a step by step explanation and a rough sketch .

Answer 1

given

the wind speed is = 10m/sec

the boat speed is = 6 m/sec

the relative speed is = ( 10 - 6 )

= 4 m/sec

for the steady state

A₁ C₁ 4² - A₂ C₂ 6² = 0

A₁ C₁ / A₂ C₂ = 36/16

A₁ C₁ / A₂ C₂ = 2.25

the new steady state speed is say V

then

new realtive spedd is = 10 - V

and the boat speed is = V

and the area is A₁' = 2 A₁

then again,

for the steady state

2 A₁ C₁ ( 10 - V )² - A₂ C₂ V² = 0

( 10 - V / V )² = A₂C₂ / 2 A₁ C₁

we have from above A₁ C₁ / A₂ C₂ = 2.25

now A₂ C₂ / A₁ C₁ = 1/2.25

( 10 - V / V )² = 1/ 2 X 2.25

( 10 - V / V )² = 1/ 4.5

10 - V / V = ( 1/4.5 )^1/2

10 - V / V = 0.471

10/V = 1 + 0.471

10/V = 1.471

V = 10/1.471

V = 6.798 m/sec

then the new steady state speed is say V = 6.798 m/sec