Question

A man is swimming in a lake in a direction of ${30}^o$ East of North with a speed of $5$ $km/hr$ and a cyclist is going on a road along the lake shore towards East at a speed of $10$ $km/hr$. In what direction and with what speed would the man appear to swim to the cyclist.

Answer 1

Refer IMAGE $01$
$|V_c|=10$ $km/hr$
$|V_s|=5$ $km/hr$
$\therefore V_s=\frac{5}{2}\uparrow +\frac{5\sqrt{3}}{2}\uparrow$
$V_c=10\uparrow$
$\therefore V_{s}c=V_s-V_c=(\frac{5}{2}\uparrow +\frac{5\sqrt{3}}{2}\uparrow )-10\uparrow =-\frac{15}{2}\uparrow +\frac{5\sqrt{3}}{2}\uparrow$
$|V_{s}c|=\sqrt{\frac{225}{4}+\frac{75}{4}}=5\sqrt{3}$ $km/hr$
Refer IMAGE $02$
$\tan \alpha =\frac{15/2}{5\sqrt{3}/2}=\sqrt{3}$
$\alpha ={60}^o$
$\therefore {60}^o$ west of north