Question

pts) boat of mass m is vo o on a lake at x0 0 m. 2. (20 A at 1 s. The boat is slowed by the water by the fo F(v) is where a a constant. Assume the boat travels in the tx direction and that this is the only force acting on it. horizontally on the a. Write the equation of motion for the boat traveling water b. Find v(t) for the boat.

Answer 1

2.a)Force, F(v)=-av^1/2

We know that force= mass * acceleration.

Let the acceleration be A.

Therefore, F(v)=mA=- av^1/2

                      mA=- av^1/2

                        A=(-av^1/2)/m

Acceleration can be expressed as

                          $\mathbf{A=v\frac{\mathrm{d} v}{\mathrm{d} x}}$

                       $\mathbf{v\frac{\mathrm{d} v}{\mathrm{d} x}=-av^{1/2}/m}$

                      $\mathbf{v^{1/2}dv=-\frac{adx}{m}}$

     Integrating both side,

                    $\mathbf{2/3v^{3/2 }-2/3v_{o}^{3/2}=-\frac{ax}{m}}$

                  $\mathbf{x=-\frac{m}{a}(2/3v^{3/2 }-2/3v_{o}^{3/2})}$

This is the equation of motion that we get.

2.b)

    Acceleration is also defined as

                                                           $\mathbf{A=\frac{\mathrm{d} v}{\mathrm{d} t}}$

                                                            $\mathbf{\frac{\mathrm{d} v}{\mathrm{d} t}=-av^{1/2}/m}$

                                                            $\mathbf{\frac{dv}{v^{1/2}}=\frac{-a}{m}dt}$

                                              Integrating both side,

                                          $\mathbf{2v^{1/2}-2v_{o}^{1/2}=\frac{-at}{m}}$

                                          $\mathbf{v =(v_{o}^{1/2}-\frac{at}{2m}})^{2}$