A ship going upstream and experiences a frictional force

[fara0815]

Hello there!

This problem is giving me a hard time and I thought maybe one of you could give me a hint:

"A ship is going upstream with a constant power P. Its speed $$V$$ is relative to the water and the water's speed $$U$$ is relative to the shore. The ship needs tp overcome a frictional force of $$F_r=cv^2$$ caused by the water which depends on its relativ speed. How big does v have to be so that the ship goes from A to B with the lowest energy consumption?"

Where I would start is that the ship needs at least the power which is neccesary to stay in the same place. Which would be
$$P=F_r x U = cU^2U=cU^3$$ and than in addition some extra power to go upstream. That would be $$P=F_r x (v-u)=c(V-U)^2 x (V-U)$$. But from here, if it's really correct, I do not know how to continue :(

 

[HallsofIvy]

I assume the "x" in your formula is a multiplication sign. Don't do that, it's confusing. Just $P= F_rU$ would be sufficient.
I don't believe you can or should distinguish between the power necessary to "stand still" and the power necessary to move forward against the current. They are the same. Note also that U is speed of the boat relative to the water so it makes no sense to use V-U (which would be negative: if the boat is moving forward against the current, U> V).