Boat crossing a river problem

Question

A ferry boat crosses a river and arrives at a point on the opposite bank directly across from its starting point. The boat can travel at 6m/s and the current is 2m/s. If the river is 700m wide, in what direction must the boat steer and how long will it take to cross?

This is exactly how the question is worded in my textbook. In "Applications of the dot and cross products" and I'm a little stumped by this one because there is no indication of what direction the current is flowing. So trying to figure out a specific direction is what's stumping me here. Also having some trouble determining the time to cross.

Answer 1

Obviously , the current has to flow against the boat to make it arrive at a point exactly opposite to the starting position.

See the Vector Triangle Diagram here

$\vec vel_{boat,ground}= \vec vel_{boat,river} + \vec vel_{river,ground}$

You need to solve the above vector equation using the triangle law.

You know that $\vec vel_{boat,ground}$ is at angle of 90° to the direction of the $\vec vel_{river,ground}$

The angle of tilt will then be: $cos^{-1}(1/3) = 70.53°$ (approx.)

For the magnitude of the $vel_{boat,ground}$, you apply the Pythogoras Theorem to the same triangle:

$vel_{boat,ground}^2= vel_{boat, river}^2 - vel_{river,ground}^2$

Or

$vel_{boat,ground}^2 = 6^2 - 2^2$

Or

$vel_{ boat,ground}$ = 5.656 m/sec.

The time is then simply

$700/5.656$

or 123.76 sec.

Answer 2

It seems quite logical to assume that the current is flowing parallel to the banks, which are two parallel straight lines, and the text indicates the crossing has to be perpendicular to those lines.

You are supposed to assume that the two movements and so the two velocity vectors, the one coming from the boat's engines and the one from the current, simply superimpose : $\vec{v} = \vec{v}_e + \vec{v}_c$.

That is, in 1 s, your boat movements will be 6 m along its steering direction plus 2 m along the flow direction. The first part of the question is to determine the steering direction so that these two motions combine into a line perpendicular to the flow direction (which corresponds to crossing the river to go to the point directly across). So because $\vec{v}$ needs to be perpendicular to the flow, $\vec{v} \cdot {\vec{v}}_c = 0$. Replacing $\vec{v}$ by the sum above, and knowing the magnitude of $\vec{v}_e$, you can find its angle with the flow.

Then you'll get also how far you get in 1 s, and you'll be able to calculate how much time is needed to cross. Since $\vec{v}$ is aligned in the correct direction, you can do this by calculating its magnitude, which is the speed, and then simply dividing distance by speed. Alternatively, you could use cross product for this, by first calculating the unit vector along the flow, $\vec{c}=\vec{v}_c/|\vec{v}_c|$, and then taking $\vec{v}\times\vec{c}$ (again by replacing $\vec{v}$) : the magnitude of this gives you the speed.