Question

1. A boat heads directly across a river with a

velocity of 12 m/s. If the river flows at 6.0 m/s find

the magnitude and direction of the boat's

resultant velocity. (State the direction relative to

an imaginary line drawn straight across the river​

Answer 1

Given:

Velocity of the boat = 12 m/s

Velocity of the river = 6 m/s

To find:

The magnitude and direction of the resultant boat‘s velocity

Calculation:

Let us assume the river is flowing towards west direction and the boat is travelling towards north direction

Velocity of the river  $=6\hat{i}$

Velocity of the boat  $=12\hat{j}$

Resultant velocity  $=6\hat{i}+12\hat{j}$

Magnitude of the boat‘s resultant velocity is given by

   $V=\sqrt{12^2+6^2}$

   $V=\sqrt{180}$

   $V=13.4\ m/s$

Direction of the boat‘s resultant velocity is given by

   $Tan\theta=\frac{b}{a} =\frac{12}{6}$

   $Tan\theta = 2$

    $\theta=63.4^{\circ}$

Final answer:

The magnitude of the boat’s velocity is 13.4 m/s and the resultant velocity makes an angle of 63.4° with the vertical

   

   

Answer 2

Answer: Magnitude = 13.4 , Direction = 26.5

Explanation:

Let us assume boat is moving in north ↑ and river is flowing in west ← direction

               Using Pythagoras  Theorem:-

                  r=$\sqrt{12^{2}+6^{2} }$    

                  r=$\sqrt{144 + 36}$

                  r=$\sqrt{180}$

                  r=13.4

Direction = tanФ = $\frac{opposite}{adjacent}$    

                   tanФ = $\frac{6}{12}$        

                   tanФ = 0.5

                      Ф = $tan^{-1}$   (0.5)

                         = 26.5

I hope it will help you :)