Waterboat Question -- which heading to take....

[isthistakenyet]

Homework Statement

If you were trying to cross a river with the shortest possible time, would you aim your boat slightly upstream, directly across the river, or slightly downstream? Explain.

Homework Equations

w = water
s = shore
b = boat

The Attempt at a Solution

I thought about this question but I'm not sure if I should take distance into account. Because, technically, while the velocity for downstream is the greatest, doesn't the distance increase? As you can see in the picture, it looks like they will all take the same amount of time because the distance keeps increasing.

 

[berkeman]

Homework Statement

If you were trying to cross a river with the shortest possible time, would you aim your boat slightly upstream, directly across the river, or slightly downstream? Explain.

Homework Equations

w = water
s = shore
b = boat

The Attempt at a Solution

I thought about this question but I'm not sure if I should take distance into account. Because, technically, while the velocity for downstream is the greatest, doesn't the distance increase? As you can see in the picture, it looks like they will all take the same amount of time because the distance keeps increasing.

Welcome to the PF.

The problem asks for the path with the shortest possible time shore-to-shore, not caring about the distance traveled. One important Relevant Equation is for the component of velocity in the horizontal direction shore-to-shore. How is that component of velocity related to the absolute velocity of the boat and the angle it is inclined above or below the horizontal?

 

[isthistakenyet]

Welcome to the PF.

The problem asks for the path with the shortest possible time shore-to-shore, not caring about the distance traveled. One important Relevant Equation is for the component of velocity in the horizontal direction shore-to-shore. How is that component of velocity related to the absolute velocity of the boat and the angle it is inclined above or below the horizontal?

But wouldn't the distance traveled increase the time it takes?

 

[berkeman]

But wouldn't the distance traveled increase the time it takes?

Write out the equation I alluded to, and you can then tell me...

 

[isthistakenyet]

Write out the equation I alluded to, and you can then tell me...

but how exactly would the component affect the time it takes, sorry I'm not very good at physics. And how do I write an equation for the component if I haven't been given any info? sorry for noobyness.

 

[berkeman]

but how exactly would the component affect the time it takes, sorry I'm not very good at physics. And how do I write an equation for the component if I haven't been given any info? sorry for noobyness.

No need to be sorry. Call the speed of the boat V. If the boat aims directly across the river (call it horizontal or x on the drawing), then Vx = V.

But if the boat aims above or below the horizontal by an angle θ then you should be able to write equations for Vx and Vy in terms of V and a trig function of θ. What are Vx and Vy in terms of θ?

 

[DaveC426913]

This is a trick question, is it not?

 

[rcgldr]

Note that the boat's "speed" is it's velocity with respect to the water that it travels across. If the boat's "speed" is zero, then it moves downstream at the same speed as the water (ignoring issues like drag from the air).

 

[haruspex]

Another approach is to use the reference frame of the water.

 

[berkeman]

This is a trick question, is it not?

Without giving away the solution, can you say why it seems like a trick question?

 

[DaveC426913]

Without giving away the solution, can you say why it seems like a trick question?

Well, I think almost everyone seems to be in agreement that the answer is simpler than it seems.

 

[SammyS]

Well, I think almost everyone seems to be in agreement that the answer is simpler than it seems.

So far, only OP seems to have a problem getting the solution.

 

[isthistakenyet]

Well, the answer turned out to be straight across :/

 

[berkeman]

Well, the answer turned out to be straight across :/

Yup. If you re-read all of the help in this thread, you'll see how we tried to help you see that on your own. Thread locked.