Question

A boat sails $2km$ east, then $4km$ southeast and then in an unknown direction. Final position of the boat is $5km$ east from starting point. Unknown displacement is:

• A. $2.8km,3^{\circ }{26}^{\prime }$ with north towards east
• B. $3km,2^{\circ }{26}^{\prime }$ with east
• C. $3.5km,2^{\circ }{30}^{\prime }$ with south towards west
• D. $1.8km,2^{\circ }{36}^{\prime }$ with north towards east

Answer 1

The correct option is A $2.8km,3^{\circ }{26}^{\prime }$ with north towards east

$\overrightarrow{S}=\overrightarrow{S_1}+\overrightarrow{S_2}+\overrightarrow{S_3}$
$5\hat{i}=2\hat{i}+\left[\right(4\cos {45}^{\circ })\hat{i}-(4\sin {45}^{\circ }\left)\hat{j}\right]+\overrightarrow{S_3}$
i.e $\overrightarrow{S_3}=(3\hat{i}-2\sqrt{2})\hat{i}+2\sqrt{2}\hat{j}$
$=0.172\hat{i}+2.828\hat{j}$
i.e $\overrightarrow{S_3}=\sqrt{{0.172}^2+{2.828}^2}$
$=2.8km$
$\tan \theta =\frac{0.172}{2.828}=0.0608$
$\theta ={\tan }^{-1}0.0608$
$\theta =3^{\circ }{26}^{\prime }$