Q:

A boat heads north across a river at a rate of 2 miles per hour. If the current is flow A boat heads north across a river at a rate of 2 miles per hour. If the current is flowing east at a rate of 5 miles per hour, find the resultant velocity of the boat. (Assume that east lies in the direction of the positive x-axis and north in the direction of the positive y-axis.)

Accepted Solution

A:
Answer:[tex]\textrm{Resultant velocity}\ =\ \sqrt{29}\ miles/hour[/tex]along the direction 68.19° from north.Step-by-step explanation:Given,speed of the boat, u= 2 miles/hour along northspeed of the river, v= 5 miles/ hour along eastSince, north and east are perpendicular to each other, so we can write the resultant velocity in vector form as,[tex]\vec{r}\ =\ 2\hat{i}+5\hat{j}[/tex]Hence, the magnitude of resultant velocity can be written as[tex]r\ =\ \sqrt{2^2+5^2}[/tex]   [tex]=\ \sqrt{4+25}[/tex]   [tex]=\ \sqrt{29}[/tex]Hence, the magnitude of resultant vector is \sqrt{29} moles/hour.And the direction of the boat can be given by,[tex]tan\theta\ =\ \dfrac{5}{2}[/tex][tex]=>\ tan\theta\ =\ 2.5[/tex][tex]=>\ \theta\ =\ tan^{-1}2.5[/tex]                  = 68.19°Hence, the resultant velocity of boat is [tex]\sqrt{29}[/tex] miles/hour along the direction making an angle 68.19° with the north.