How Fast is the River Flowing in the Fisherman's Problem?

[PPonte]

Problem:

A fisherman sets out upstream from Metaline Falls on the
Pend Oreille River in northwestern Washington State. His
small boat, powered by an outboard motor, travels at a
constant speed v in still water. The water flows at a lower
constant speed vw. He has traveled upstream for 2.00 km
when his ice chest falls out of the boat. He notices that
the chest is missing only after he has gone upstream for
another 15.0 minutes. At that point he turns around and
heads back downstream, all the time traveling at the same
speed relative to the water. He catches up with the floating
ice chest just as it is about to go over the falls at his
starting point. How fast is the river flowing? Solve this
problem in two ways. (a) First, use the Earth as a reference
frame. With respect to the Earth, the boat travels upstream
at speed v - v_w and downstream at v + v_w. (b) A
second much simpler and more elegant solution is obtained
by using the water as the reference frame. This approach
has important applications in many more complicated
problems; examples are calculating the motion of
rockets and satellites and analyzing the scattering of subatomic
particles from massive targets.​

I had no problem solving (a), my doubts arise when solving (b).

I desire to know if the following considerations for (b) are right:

• When the boat is traveling upstream his speed relative to the water reference frame is v + v_w or simply v?
• When the boat is traveling downstream his velocity relative to the water reference frame is v - v_w or simply v?
• The chest is motionless relative to the water reference frame.
  
  
  

Thanks in advance for your help.

 

[Hootenanny]

I desire to know if the following considerations for (b) are right:

• When the boat is traveling upstream his speed relative to the water reference frame is v + v_w or simply v?
• When the boat is traveling downstream his velocity relative to the water reference frame is v - v_w or simply v?
• The chest is motionless relative to the water reference frame.

Hi there PPonte,

I've not been on the forums for a while, but it's nice to 'speak' to you again. All your considerations seem valid to me.

Edit: My apologies PPonte, I didn't see that your first two statements were actually questions. As SGT says, the velocity of the boat in the water's frame of reference is always v.

 

[SGT]

The velocity of the boat relative to the water is v, as is stated in the problem, no matter if the boat is traveling up or downstream.