Answer to Question #120885 in Mechanics | Relativity for Shishir mondal

Question

A scouting party has become trapped in a forest away from their field camp. Based on their explorations, the scouts know that they are 2 km from the field camp in a direction 30˚ west of north. They also know that field camp is located 3 km from the base camp in a direction 45˚ north of east. They wish to radio their position to the base camp so that food and supplies can be dropped by air as close to their position as possible. How can they pinpoint their location relative to the base camp?

Answer 1

Let, assume the base camp is at origin $A=(0,0)$

Now the rough sketch of the scenario will be

Let,the position of field camp is at $C$ ,

Thus, we have

$||\overrightarrow{AB}||=||\overrightarrow{u}||=3\:km\\ ||\overrightarrow{BC}||=||\overrightarrow{v}||=2\:km$

Clearly from the above figure

$C=(3\cos(45^{\circ})-2\sin(30^{\circ}),3\sin(45^{\circ})+2\cos(30^{\circ}))\\ \implies C=(\frac{3}{\sqrt{2}}-1,\frac{3}{\sqrt{2}}+\sqrt{3})=(1.12,3.85)$

Thus,

$||\overrightarrow{AC}||=||\overrightarrow{w}||=\sqrt{1.12^2+3.85^2}\approx 4\:km$

And direction is

$\theta=\tan^{-1}(\frac{1.12}{3.85})=\tan^{-1}(\frac{1.12}{3.85})=16.22^{\circ}$

Along north of east.

Therefore, the position of field camp from base camp is $4\:km$ along $16.22^{\circ}$ north of east