Vectors Question using Calculus -- Swimmer crossing a River

[ttpp1124]

Homework Statement

Vectors Question

Relevant Equations

Not Available

Hi! I have a physics question I need help with.

Bob can swim at 4 m/s in still water. He wishes to swim across a river 200 m wide to a point directly opposite from where he is standing. The river flows westward at 2.5 m/s and he is standing on the South bank of the river.

a. What is the speed of Bob relative to the ground?

b. In what direction Bob must head? How much time will it take?

 

[kuruman]

To get help you have to make an effort to answer the question and show what it is.

 

[ttpp1124]

Will I be using Pythagorean's formula?

 

[kuruman]

Use it, post it and explain what you think it means.

 

[ttpp1124]

I'm not sure if this is right.. if it is, I am thinking of a way to calculate the direction.

 

[kuruman]

Why does this give you the answer you are looking for? Remember the idea is that Bob reaches a point directly across. How is that figured in your equation? Look at part (b). Must Bob aim himself straight across in order to get to a point straight across?

 

[ttpp1124]

He's swimming straight ahead. He's moving in a straight line, so you're right we wouldn't use that. Would I move on directly to an equation that solves for speed?

 

[kuruman]

You have to devise the equation. Reason it out. Draw a diagram of Bob's velocity vector relative to the straight across direction. You already know that it can't point straight across. In what direction must it point?

 

[ttpp1124]

From what I acknowledge, he's moving perpendicular to the current. Is he going North?

 

[kuruman]

The current flows east to west. If he swims directly to the north, will he end up at a point directly across?

 

[ttpp1124]

No, he will need to swim west

 

[kuruman]

You mean with the current which also flows west?

 

[ttpp1124]

Yes. He can't be going north and the question doesn't say anything about him moving against the current.

 

[kuruman]

Yes. He can't be going north and the question doesn't say anything about him moving against the current.

It doesn't say anything about him moving with the current either. It does say that he wants to end up straight across. That's what counts. Does this mean that he has to angle himself with the current or against the current?

 

[PeroK]

Bob can swim at 4 m/s in still water.

I often wonder about the people who come up with these questions. The fastest that any human can swin is about $2m/s$. Bob could swim the $100m$ in about $25s$, which is half the current world record.

I know it's not relevant to the problem, but I thought it was worth pointing out.

 

[Master1022]

Homework Statement:: Vectors Question
Relevant Equations:: Not Available

Hi! I have a physics question I need help with.

Bob can swim at 4 m/s in still water. He wishes to swim across a river 200 m wide to a point directly opposite from where he is standing. The river flows westward at 2.5 m/s and he is standing on the South bank of the river.

a. What is the speed of Bob relative to the ground?

b. In what direction Bob must head? How much time will it take?

Just a tip with these sorts of relative velocity/ closest approach questions, you should ALWAYS draw out a diagram (as suggested by @PeroK ). We know that: \vec v_{B} = \vec v_{W} + \vec v_{B/W}, where W represents the water, B is Bob, and B/W is the velocity of Bob relative to the water. You want to construct a vector triangle such that you can choose the direction of \vec v_{B}

(If you still need some help/ practice with the geometric methods for these sorts of questions, I would have a look at the Edexcel M4 textbook in this google drive: https://drive.google.com/drive/folders/0B1ZiqBksUHNYX3dkQXFRQ0NlNDA - you might have to download it, but there is plenty of practice in that book on exactly these sorts of problems).

Hope that is of some help.

 

[ttpp1124]

Just a tip with these sorts of relative velocity/ closest approach questions, you should ALWAYS draw out a diagram (as suggested by @PeroK ). We know that: \vec v_{B} = \vec v_{W} + \vec v_{B/W}, where W represents the water, B is Bob, and B/W is the velocity of Bob relative to the water. You want to construct a vector triangle such that you can choose the direction of \vec v_{B}

(If you still need some help/ practice with the geometric methods for these sorts of questions, I would have a look at the Edexcel M4 textbook in this google drive: https://drive.google.com/drive/folders/0B1ZiqBksUHNYX3dkQXFRQ0NlNDA - you might have to download it, but there is plenty of practice in that book on exactly these sorts of problems).

Hope that is of some help.

If I'm drawing a triangle, the hypotenuse would be B/W, right? I think that the adjacent side would be Bob and the opposite side would be the water.

 

[PeroK]

If I'm drawing a triangle, the hypotenuse would be B/W, right? I think that the adjacent side would be Bob and the opposite side would be the water.

One way to think about this is to consider the two motions separately: Bob swimming in the water; and the water moving. Imagine Bob swims for $1s$, like a dolphin(!), and moves $4m$. Then, imagine the water moves for $1s$: that's $2.5m$. When you put these two together that's where Bob ends up, after $1s$.

That's one way to explain why he can't just aim directly across at the opposite bank. He would swim $4m$ across and then the water would take him $2.5$ downstream, and he wouldn't be going in the direction he wants towards the opposite bank.

Is B/W the hypoteneuse? Is it necessarily a right-angle triangle?

 

[ttpp1124]

One way to think about this is to consider the two motions separately: Bob swimming in the water; and the water moving. Imagine Bob swims for $1s$, like a dolphin(!), and moves $4m$. Then, imagine the water moves for $1s$: that's $2.5m$. When you put these two together that's where Bob ends up, after $1s$.

That's one way to explain why he can't just aim directly across at the opposite bank. He would swim $4m$ across and then the water would take him $2.5$ downstream, and he wouldn't be going in the direction he wants towards the opposite bank.

Is B/W the hypoteneuse?

Since he can't swim directly across, he would have to swim in two different directions.

 

[kuruman]

Since he can't swim directly across, he would have to swim in two different directions.

What does that mean? Can you walk in two different directions? There is only of you moving. You are on the right track (no pun intended) but can you say what you mean in terms of a vector and its components?

 

[ttpp1124]

What does that mean? Can you walk in two different directions? There is only of you moving. You are on the right track (no pun intended) but can you say what you mean in terms of a vector and its components?

 

[ttpp1124]

Is the answer above correct...?

 

[PeroK]

View attachment 258488

That shows what would happen if Bob aimed directly across at the opposite bank and got taken downstream with the river. That's not what you (or Bob) wants!

 

[ttpp1124]

Is this better?

 

[PeroK]

View attachment 258490
Is this better?

Who says it must be a right-angle triangle?

 

[jbriggs444]

View attachment 258490
Is this better?

That's the same drawing turned upside down.

You want the sum of the river velocity and the swimmer velocity to give a result that points directly across the river. Can you draw a right triangle to show that?

The flow velocity and the desired velocity are at right angles. If those are two of the sides of the triangle, it will indeed be a right triangle.

 

[PeroK]

Who says it must be a right-angle triangle?

I meant, of course, why does the direction Bob's swims have to be at a right angle to the current?

 

[ttpp1124]

A lot of my problems have right-angled triangles with them..
Is it not supposed to be a right angle?

 

[PeroK]

A lot of my problems have right-angled triangles with them..
Is it not supposed to be a right angle?

As @jbriggs444 pointed out, it's the river's velocity and Bob's resultant velocity that you know are at right angles. Bob can swim at angle any relative to the river.

 

[ttpp1124]

H

I meant, of course, why does the direction Bob's swims have to be at a right angle to the current?

He's swimming to the oppiste side.

As @jbriggs444 pointed out, it's the river's velocity and Bob's resultant velocity that you know are at right angles. Bob can swim at angle any relative to the river.

Relative meaning in the direction of.
I drew another diagram, I'm not sure if I'm on the right track..

 

[etotheipi]

Relative meaning in the direction of.

I know this is an old thread, but this just bothered me. Relative to does not mean 'in the direction of', relative to implies a subtraction i.e. 'from the perspective of'. A part of a vector in a certain direction is something different, i.e. a vector component / vector projection.