A fisherman sets out upstream on a river. His small boat, powered by an outboard

Question

A fisherman sets out upstream on a river. 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. The fisherman has traveled upstream for 2.37 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 17.6 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 he returns to his starting point. How fast is the river flowing?
Find the answer by using the water as reference frame . (the answer is 4.04 i found it by using earth as reference frame but the question also wants this way and I couldnt find it )

Explanation / Answer

The ice chest floats in the downstream motion 2.37km in time t,

so that 2.37 km = vwt.
The upstream motion of the boat is d = (v-vw)17.6 min.
The downstream motion is: d + 2.37 km = (v+vw)17.6 min.
t = 2.37 km/vw
substituting this value,
(v-vw)17.6 min + 2.37 km = (v+vw)(2.37 km/vw - 17.6 min)
v(17.6 min) - vw(17.6 min) + 2.37 km = v/vw(2.37km) + 2.37km -v(17.6 min) - vw(17.6 min)
v(35.2 min) = v/vw(2 km)
vw = 2.37 km/ 35.2 min
     = 4.0397 km/h
This is the velocity of water with respect to the ground.
In the reference fram of the water, the chest is not moving. The boat travels upstream for 17.6 min at speed v, and then downstream at the same speed, to return to the same point.
So, it travels for 35.2 min. During this time, the falls approach the chest at speed vw, traveling 2.37 km. So,
vw= x/t
= 2.37km/35.2min
= 4.0397 km/h

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