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).

 

[kyle8921]

I can definitely understand how this problem is giving you a hard time. It seems like a complex and multi-dimensional problem. To start, we can break down the forces acting on the ship into two components: the driving force (P) and the resisting force (F_r). The driving force is responsible for propelling the ship forward, while the resisting force is caused by the friction between the ship and the water.

To determine the minimum energy consumption for the ship to travel from point A to point B, we need to consider the work done by the driving and resisting forces. The work done by the driving force is simply P x distance traveled, while the work done by the resisting force is F_r x distance traveled.

Since we want to minimize the energy consumption, we want to minimize the total work done. This means that the ratio of work done by the driving force to the work done by the resisting force should be as low as possible.

Using this information, we can set up an equation: P x distance traveled / F_r x distance traveled = P / F_r. This represents the ratio of the work done by the driving force to the work done by the resisting force.

To minimize this ratio, we can take the derivative with respect to v (the speed of the ship) and set it equal to 0. This will give us the optimal speed (v) for the ship to travel from A to B with the lowest energy consumption.

I hope this helps give you a starting point for solving this problem. Remember, as a scientist, it's important to break down complex problems into smaller, more manageable parts and use equations and principles to guide your thinking. Good luck!