Question

A man rows a boat with a speed of \[24\;{\rm{km/hr}}\] \[30^\circ \] north of west direction. The shoreline makes an angle of \[15^\circ \] south of west. Obtain the component of the velocity of the boat along the shoreline is
A. \[6\sqrt 2 \;{\rm{km/h}}\]
B. \[24\cos 15^\circ \;{\rm{km/h}}\]
C. \[24\sin 15^\circ \;{\rm{km/h}}\]
D. \[12\sqrt 2 \;{\rm{km/h}}\]

Answer 1

Hint:The above problem can be resolved by using the fundamentals of vector components. In the given situation, it is said that the boat is moving with some magnitude of speed and also the motion of the boat is along the direction of north and west. And this direction is given in terms of angle and also the data regarding the shoreline is given. This shoreline is along the direction of the south-west, such that angle made by boat along the shoreline is given by taking sums of angles. And at last, the component of velocity along the shoreline will be considered by taking the horizontal component of velocity in a similar direction.

Complete step by step answer:
Given data:
The speed of the boat is, \[{v_1} = 24\;{\rm{km/hr}}\].
The angle made by the north west direction is, \[{\theta _1} = 30^\circ \].
The angle made by the south west is, \[{\theta _2} = 15^\circ \].
The angle made by the boat along the shoreline is,
\[\begin{array}{l}
\theta = {\theta _1} + {\theta _2}\\
\theta = 30^\circ + 15^\circ \\
\theta = 45^\circ
\end{array}\]
Then the component of velocity along the shoreline is,
\[v = {v_1}\cos \theta \]
Solve by substituting the values as,
 \[\begin{array}{l}
v = {v_1}\cos \theta \\
v = 24\;{\rm{km/hr}} \times \cos 45^\circ \\
v = 24\;{\rm{km/hr}} \times \left( {\dfrac{1}{{\sqrt 2 }}} \right)\\
v = 12\sqrt 2 \;{\rm{km/hr}}
\end{array}\]
 Therefore, the component of velocity along the shoreline is \[12\sqrt 2 \;{\rm{km/hr}}\] and option (D) is correct.

Note: To resolve the given problem, one must be clear about the concept of the vectors. The velocity is also considered as an important variable and is also considered as a vector quantity, that has broader significance in resolving problems of kinematics. Moreover, one must learn to determine the components of velocities and other fundamentals while resolving similar kinds of problems.