Question

a river flows due south at 1.6 mi/h, and a swimmer attempts to cross the river from the west side to the east side. in what direction should the swimmer head (in degrees e of n), at a velocity of 3 mi/h, in order to arrive at a landing point due east of the starting point? (round your answer to one decimal place.)
n
° e

Explanation:

Step1: Set up the velocity vectors

Let the velocity of the river \( \vec{v}_{r}=- 1.6\vec{j}\) (south - direction) and the velocity of the swimmer relative to the water be \( \vec{v}_{s/w}=3(\cos\theta\vec{i}+\sin\theta\vec{j})\). The resultant velocity \( \vec{v}=\vec{v}_{s/w}+\vec{v}_{r}=3\cos\theta\vec{i}+(3\sin\theta - 1.6)\vec{j}\). Since the swimmer wants to land due east, the \(y\) - component of the resultant velocity is \(0\).

Step2: Solve for \(\theta\)

Set \(3\sin\theta-1.6 = 0\). Then \(\sin\theta=\frac{1.6}{3}\approx0.533\). Using the inverse - sine function, \(\theta=\sin^{-1}(0.533)\approx32.2^{\circ}\) north of east.

Answer:

\(32.2\)