Question

A fairy carries people from a landing on one side of a river, 1.6 km, to a landing on the other. The ferry captain knows The aging ship can reach a maximum sustainable speed of 12 km/hr. During a day in the rainy season the Captain estimates the speed of the River to be 9.0 km/hr downstream. A) at what angle to the direct line of crossing must the captain steer the ferry in order to arrive on the bank directly across the river at the maximum sustainable speed? B) how long does it take to cross and get to the landing on the other side?

Answer 1

We will have the following:

A) From this we will have that the angle will be given by:

$\begin{gathered} 9km/h-(12km/h)sin(\theta)=0\Rightarrow sin(\theta)=\frac{9km/h}{12km/h} \\ \\ \Rightarrow\theta=sin^{-1}(\frac{3}{4})\Rightarrow\theta=48.59037789... \\ \\ \Rightarrow\theta\approx48.6 \end{gathered}$

So, it must point approximately 48.6° upstream.

B) We will determine the time it takes to cross to the other side as follows:

First, we determine speed used to move to the other side:

$\begin{gathered} v_y=(12km/h)cos(sin^{-1}(\frac{3}{4}))\Rightarrow v_y=7.937253933...km/h \\ \\ \Rightarrow v_y\approx7.9km/h \end{gathered}$

So, the time it will take to cross to the other side will be:

$\begin{gathered} t\approx\frac{(1.6km)(1h)}{(7.9km)}\Rightarrow t\approx\frac{19}{79}h \\ \\ \Rightarrow t\approx0.2h \end{gathered}$

So, it will take approximately 0.2 hours; that is approximately 12 minutes.