Question

A sailboat sails $2km$ due East, $5km{37}^{\circ }$ South of East and finally an unknown displacement. If the final displacement of the boat from the starting point is $6km$ due East, determine the third displacement

• A. $3km$, West
• B. $5km$, East
• C. $4km$, South
• D. $3km$, North

Answer 1

The correct option is D $3km$, North

Draw a rough diagram.

Find unknown displacement.
Given,
resultant displacement is $6km$ due east which may be expressed as $6\hat{i}$.
$2km$ due east,
${\overrightarrow{S}}_1=2\hat{i}$
$5km{37}^{\circ }$ south of east,
its components along $x-\text{axis}$ $=5\cos {37}^{\circ }=5\times \frac{4}{5}=4\hat{i}$
along $y-\text{axis}=-5\sin {37}^{\circ }=-5\times \frac{3}{5}=-3\hat{j}$
Let the third displacement be ${\overrightarrow{S}}_3.$
$\therefore 6\hat{i}=2\hat{i}+\left(5cos{37}^{\circ }\hat{j}\right)+{\overrightarrow{S}}_3$
Or
$6\hat{i}=2\hat{i}+(4\hat{i}-3\hat{j})+{\overrightarrow{S}}_3$
$\Rightarrow$ Unknown displacement, ${\overrightarrow{S}}_3=3\hat{j}$
Final Answer: (a)