Question

A river flows due south at $3mi/h$. A swimmer attempting to cross the river heads due east swimming at $2mi/h$ relative to the water. Find the true velocity of the swimmer as a vector.

Short answer

The true velocity of swimmer is $\left|W\right|\approx 3.6$

Step-by-step solution

Step 1. Given information

A river is flowing due south at $3mi/h$ and a swimmer swimming due east at $2mi/h$

Step 2. Concept used

First we have to write velocity into vector form. To do this, we need direction and the modulus of vector. A vector in component form can be written as-

$v=a_{1}i+a_{2}j$

Where $a_1=\left|v\right|\cos \theta \&a_2=\left|v\right|\sin \theta$

$a_1$ is the horizontal component and $a_2$is the vertical component. And we know that we take North is in vertical direction.

Step 3. Calculation

East component of velocity can be written as-

$U=2i$

And swimmer will have velocity in north direction also because river flows him in north direction as well.

The north component is-

$V=-3j$

True velocity is the modulus or magnitude of velocity vector,

$\begin{array}{l}\left|W\right|=\sqrt{2^2+{(-3)}^2}\\ =\sqrt{4+9}\\ =\sqrt{13}\\ =3.6055\\ \approx 3.6\end{array}$