Q3.4-21E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

A sailboat has been running (on a straight course) under a light wind at 1 m/sec. Suddenly the wind picks up, blowing hard enough to apply a constant force of 600 N to the sailboat. The only other force acting on the boat is water resistance that is proportional to the velocity of the boat. If the proportionality constant for water resistance is b = 100 N-sec/m and the mass of the sailboat is 50 kg, find the equation of motion of the sailboat. What is the limiting velocity of the sailboat under this wind?

Step-by-Step Solution

Answer

• The equation of motion of sail boat is $\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{6}\mathbf{t}\mathbf{+}\frac{\mathbf{5}}{\mathbf{2}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{-}\frac{\mathbf{5}}{\mathbf{2}}$ . 
• The limiting velocity is 6 m/sec.

Step 1: Find the equation of velocity

There are two forces are 

 

 $$\begin{gathered} \mathrm{F}=600 \\ {\mathrm{F}}_2=-100\mathrm{v} \end{gathered}$$

 

Now

  $$\begin{gathered} \mathrm{m}\frac{\mathrm{dv}}{\mathrm{dt}}=600-100\mathrm{v} \\ 50\frac{\mathrm{dv}}{\mathrm{dt}}=600-100\mathrm{v} \\ \frac{\mathrm{dv}}{\mathrm{dt}}=12-2\mathrm{v} \\ \int \frac{\mathrm{dv}}{6-\mathrm{v}}=\int 2\mathrm{dt} \\ -6 \ln \left|6-\mathrm{v}\right|=\mathrm{t}+\mathrm{C} \\ \mathrm{v}\left(\mathrm{t}\right)=6-\mathrm{C}{\mathrm{e}}^{-2\mathrm{t}} \end{gathered}$$

 

Put v(0) = 1 then C = 5

 $\mathrm{v}\left(\mathrm{t}\right)=6-5{\mathrm{e}}^{-2\mathrm{t}}$

 

Step 2: Find the value of equation of motion

$$\begin{gathered} \mathrm{x}\left(\mathrm{t}\right)=\int 6-5{\mathrm{e}}^{-2\mathrm{t}}\mathrm{dt} \\ \mathrm{x}\left(\mathrm{t}\right)=6\mathrm{t}+\frac{5{\mathrm{e}}^{-2\mathrm{t}}}{2}+\mathrm{A} \end{gathered}$$

When t = 0 then  

 $\mathrm{x}\left(\mathrm{t}\right)=6\mathrm{t}+\frac{5{\mathrm{e}}^{-2\mathrm{t}}}{2}+\frac{5}{2}$

 

Hence, the equation of motion of sail boat is  $\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{6}\mathbf{t}\mathbf{+}\frac{\mathbf{5}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}}{\mathbf{2}}\mathbf{+}\frac{\mathbf{5}}{\mathbf{2}}$.

The limiting velocity of the sailboat is 

$\begin{array}{rcl}\underset{t\to \infty }{\lim v\left(t\right)}& =& \underset{t\to \infty }{\lim } (6-5e^{-2t})\\ & =& 6\end{array}$

Hence, the Limiting velocity is 6 m/sec.