Question

A boat moves relative to water with a velocity half of the river flow velocity if the angle from the direction of flow with which boat must move relative to stream direction to minimise drift is 2 pie by n then find n

Answer 1

Final Answer : n= 3 

Solution:
 
Steps: 
1) Let the relative velocity of boat be 'v' m/s making an angle  $\theta$ with vertical .Then,the velocity of river is '2v' in hrizontal direction .
Then,we will try to break velocity of boat in two components .

2) We observe that resultant velocity of boat :
               
        $v_{x} = 2v - vsin(\theta)$ 
         $v_{y} = vcos(\theta)$
         
3) Let the width of river be 'd' . 
Time required to cross the river,t : $\frac{d}{vcos(\theta)}$ 

Since, we observed that $v_{x} \ \textgreater \ 0$ ,so drift can't be 0.

4) That is, 
     Drift ,x =   $v_{x}\times t$ 
                 = $= (2v-vsin(\theta)) \times \frac{d}{vcos(\theta)}$

For Drift to be minimum, 
$\frac{dx}{d\theta} = 0$ 
$\frac{d}{d\theta}( \frac{d}{v}(2vsec(\theta)-vtan(\theta)) ) \\ =\ \textgreater \ \frac{d}{v} (2vsec(\theta)tan(\theta)-v sec^{2}(\theta) ) =0 \\ =\ \textgreater \ 2vtan(\theta)-vsec(\theta) =0 \\ =\ \textgreater \ 2vsin(\theta) =v \\ =\ \textgreater \ sin(\theta) = \frac{1}{2} \\ =\ \textgreater \ \theta = 30\degree$

5) Hence, angle made with the vertical is 30 degree.
Therfore,angle made with the direction of river flow is 90+30=120 degree.
  
$120\degree= \frac{2\pi}{3} radian$
Hence , value of n is 3. 

Answer 2

Answer:

Explanation:here you go