How Does River Current Affect Boat Speed and Crossing Time?

[Inertialforce]

Homework Statement

A boat which can travel at 5.6 m/s in still water heads due east across a river (185m wide) from a dock at a point X. The boat's resultant path is 32(degrees) south of east.

a)What is the speed of the current?

b)How long will it take the boat to reach the far shore if the river is 185m wide?

Homework Equations

Vwg = Vwb + Vbg

The Attempt at a Solution

Here is what I did for a) :

First I wrote down what I was given:
dx = 185m
vx= Vbw = 5.6m/s
(theta) = 32(degrees)

then I went:

Tan(32) = Vwg / Vbw
Tan(32) = Vwg / 5.6
(Tan(32))(5.6) = Vwg
3.499268371 = Vwg
3.5m/s = Vwg

then for b) I went:

Vx = dx/t
t = dx/vx
t = 185/5.6
t = 33.03571429
t = 33s

could someone please check these to see if they are correct, I just want to see if I am doing it correctly before moving on to other questions.

Furthermore, could some please draw a vector diagram for "part A" so that I could understand this question better because someone told me that a vector diagram would help me to understand this question a lot more. However, in the question sheet there is already a birds eye view of what is occurring in this question already so I was just wondering what a alternative vector diagram would look like.

 

[Mentallic]

I don't think you are understanding the effects the current has on the ship crossing the river. You were told the ship can travel at 5.6ms^-1 on its own. But when it tries to travel straight across the river, it is being dragged south with the current so it makes an angle of 32^o south from the east.
Now you can make a right triangle of all the vectors. i.e. the 5.6ms^-1 speed of the ship as the hypotenuse, the current speed being opposite to the dock X and the speed that will be made to cross the river.

 

[Carid]

When I have watched ferries cross a river they tend to head upstream at the angle necessary to arrive at a point exactly opposite the starting point. That however is not what the question is asking. Surely if the boat heads straight across the river it will do so at a speed of 5.6m/sec. The current which is a right angles to the longitudinal axis of the boat sweeps the boat downstream at the speed of the current (sc). So we have a right angle triangle in which for a given second of travel :-

5.6/sin(90°-32°)=sc/sin(32°)
sc = 5.6 * sin(32°)/sin(58°)
sc=3.5m/sec

Now the downstream movement induced by the current should not influence the crossing time which should still be 185m/5.6ms-1 = 33sec or thereabouts.

The original answers look correct.