Crosswind problem (pgs. 34-35, Thinking Physics, 3rd edition)

[A.T.]

Maximizing $v_x$ maximizes $v$. The motion is constrained by $v_y = v_x \tan \theta$

I was replying this part:

... by maximizing the force $F_x$.

Maximizing $F_x$ doesn't maximize $v$ or $v_x$. For example: If $F$ is perpendicular to the track, then no matter how large you make $F$ and $F_x$ it won't start moving.

 

[erobz]

I was replying this part:

Maximizing $F_x$ doesn't maximize $v$ or $v_x$. For example: If $F$ is perpendicular to the track, then no matter how large you make $F$ and $F_x$ it won't start moving.

Ok, I see the issue.

 

[A.T.]

I don't understand it.

Just plug in some numbers into your solution to see if it makes any sense.

$$\lim_{\dot{v}_x \to 0 }(5) \implies v_x \to -w\frac{( \tan \theta - \cos \beta \tan \theta - \sin \beta)}{( 1 + \tan^2 \theta )}$$

My position is that this result does not support the claim that $v_x$ can exceed $w$.

The angles in the image below are roughly $\theta=33°$ and $\beta = 43°$. When you plug these values into your solution you get a limit for $v_x$ of about $0.36 w$.

If the track is completely frictionless, why would the below cart (even with the fixed vane) stop accelerating at $v_x = 0.36w$?

The incoming relative flow w' would be ~20° clockwise from the x-axis, blowing right onto the vane, creating a force F roughly as shown, which only needs to have some non-zero component in the direction of v for acceleration.

 

[erobz]

Just plug in some numbers into your solution to see if it makes any sense.The angles in the image below are roughly $\theta=33°$ and $\beta = 43°$. When you plug these values into your solution you get a limit for $v_x$ of about $0.36 w$.

If the track is completely frictionless, why would the below cart (even with the fixed vane) stop accelerating at $v_x = 0.36w$?

The incoming relative flow w' would be ~20° clockwise from the x-axis, blowing right onto the vane, creating a force F roughly as shown, which only needs to have some non-zero component in the direction of v for acceleration.

View attachment 320802

$w$ is not in the direction you show?

 

[A.T.]

$w$ is not in the direction you show?

$w$ : true wind, air motion relative to the ground, is along positive x direction
$w'$ : apparent wind, air motion relative to cart, is given by the vector equation:

$w' = w - v$ (see also vector diagram top-left)

 

[erobz]

$w$ : true wind, air motion relative to the ground, is along positive x direction
$w'$ : apparant wind, ari motion relative to cart, is given by the vector equation:

$w' = w - v$ (see also vector diagram top-left)

View attachment 320808

No the true wind $w$ is in $-y$ direction for the analysis you are quoting.

 

[A.T.]

No the true wind $w$ is in $-y$ direction for the analysis you are quoting.

That doesn't make any sense. Why are you solving for the limit of $v_x$ if true wind is along $-y$. The whole dispute was about the limit of velocity component parallel to true wind.

 

[erobz]

That doesn't make any sense. Why are you solving for the limit of $v_x$ if true wind is along $-y$. The whole dispute was about the limit of velocity component parallel to true wind.

Because you were asking me to flip flop all the directions of things, the sail, the angle of the vane and the wind! I had said this doesn't make sense for the question a long time ago. see post #109, and prior to that post #101. I brought up this point several times now.

The fluid jet is the true wind. What configuration do you want me to solve?

 

[A.T.]

Because you were asking me to flip flop all the directions of things, the sail, the angle of the vane and the wind!

I never told you to change the true wind from positive x. I told you to compute the wind relative to the vane, which for $v_x = w$ is indeed along negative y.

The fluid jet is the true wind.

There is no jet. The moving airmass is continuous. Your jet is just a source of confusion you have introduced for yourself. Please learn how to add vectors and transform velocities between frames. I have posted the correct vector math many times.

 

[erobz]

I never told you to change the true wind from positive x. I told you to compute the wind relative to the vane, which for $v_x = w$ is indeed along negative y.There is no jet. The moving airmass is continuous. Your jet is just a source of confusion you have introduced for yourself. Please learn how to add vectors and transform velocities between frames. I have posted the correct vector math many times.

No, you're not understanding the intentions. The jet(s) - an array of what is shown are the are the wind!

We think about the wind flow as lamina (the individual jets impacting the sail as the cart moves along the track), in the limit that the distance between the jets goes to zero.

 

[A.T.]

the individual jets impacting the sail

The fluid jet is the true wind.

The sail experiences the apparent wind (relative to the boat), not the true wind. If your jets are the true wind (relative to the ground) then the interaction between jet and sail in your image is nonsense. This is how the interaction between sail and apparent wind looks like.

Forget your jets. Just do the proper vector math.

 

[erobz]

The sail experiences the apparent wind (relative to the boat), not the true wind. If your jets are the true wind (relative to the ground) then the interaction between jet and sail in your image is nonsense. This is how the interaction between sail and apparent wind looks like.

Forget your jets. Just do the proper vector math.

Well, correct it within the appropriate framework then i.e. Newtonian Mechanics. That's how we do Classical Mechanics. We don't add randomly drawn vectors and say, "look at the way they point, that proves it..."

OR Just forget about it and go enjoy your faster than wind down wind sailing. I'm through wasting any more time on this back and forth, there is no ground to gain.

Take Care!

 

[A.T.]

Well, correct it within the appropriate framework then i.e. Newtonian Mechanics.

I did correct your errors, as far I could understand what you are trying to do. But if you can't even keep true and relative wind apart, because you insist on thinking in terms of "jets", then there is not much I can do.

In the end, the much simpler vector approach agrees with empirical evidence. Your "jets" don't. And that's all that matters in physics.

 

[A.T.]

We don't add randomly drawn vectors and say, "look at the way they point, that proves it..."

If those vectors look "random" to you, then I can recommend some reading that explains them further:

High-speed sailing, Wolfgang Püschl 2018 Eur. J. Phys. 39 044002
https://iopscience.iop.org/article/10.1088/1361-6404/aab982

Physics of Sailing, John Kimball
https://books.google.de/books?id=Xe_i23UL4sAC&lpg=PP1&hl=de&pg=PA49#v=onepage&q&f=false

Course Theorem, Lester Gilbert
https://www.onemetre.net//design/CourseTheorem/CourseTheorem.htm

 

[A.T.]

What configuration do you want me to solve?

This is the situation at $v_x = w$:

$w$: true wind (relative to the ground) along positive x
$v$: velocity of the cart relative to the ground
$w'_{in}$: incoming apparent wind (relative to the vane) along negative y (vector equation: $w'_{in} = w - v$)
$w'_{out}$: deflected apparent wind (relative to the vane)
$F$: force on the vane (vector equation: $F = \dot{m}w'_{in} - \dot{m}w'_{out}$)
$F_v$: component of $F$ parallel to $v$ (vector equation: $F_v = F \cdot \hat{v}$)

As long as $F_v$ is positive, we can accelerate further, and $v_x = w$ is not a limit.

 

[Gleb1964]

There is no jet. The moving airmass is continuous.

I would agree even with unrealistic "jet wind" concept, asking consider infinity long sail instead.
Your explanations are good.
Once in past disputing the same subject (if its possible to descent faster than the wind speed), I did a picture with the boat on a flow. Using a lever connected to the ground, the boat is going faster than the flow. The boat is using the speed difference between flow and ground, exchanging momentum both with flow and ground.

 

[A.T.]

I would agree even with unrealistic "jet wind" concept, asking consider infinity long sail instead.

The problem with @erobz 's jet model is that causes him to confuse true and relative wind. He makes his jets parallel to the true wind, and then seems to falsely assume that these are the stream-lines of the relative wind that the airfoil interacts with. But the relative wind has a completely different direction when you move across the true wind.

Once in past disputing the same subject (if its possible to descent faster than the wind speed), I did a picture with the boat on a flow. Using a lever connected to the ground, the boat is going faster than the flow. The boat is using the speed difference between flow and ground, exchanging momentum both with flow and ground.

Your image is about going directly downwind faster than the wind, which we even didn't get into here. Some similar animations:

 

[Gleb1964]

Your image is about going directly downwind faster than the wind, which we even didn't get into here.

Descending directly downwind faster than wind would need lateral speed component for sail, like making spinning propeller sail, where propeller rotation forced by motion relative to water.

The problem with @erobz 's jet model is that causes him to confuse true and relative wind. He makes his jets parallel to the true wind, and then seems to falsely assume that these are the stream-lines of the relative wind that the airfoil interacts with.

Perhaps, the sailing boat can be considered in any frame with the same outcome.
I would take the model of boat going downwind with the speed many times exceed the wind speed to get it the obvious that the apparent wind would be filling like counter wind moving in any directions. But nevertheless, in the most directions it is possible to find a vane position when sail is getting a positive force, keeping boat moving. The boat can descend faster than wind, but the sail surface can descend slower than wind, taking wind momentum into the sail.

 

[A.T.]

Perhaps, the sailing boat can be considered in any frame with the same outcome.

The change in the air's momentum (and thus the force) is frame independent. But the direction of the incoming and outgoing flows are frame dependent. In the image below the blue/purple line is the stream line in the ground frame (similar to @erobz jets). But the stream relative to the sail is indicated by the red dots.

Here it is animated:

 

[erobz]

Same as for a boat with a fixed sail going directly downwind. But the boats that achieve downwind components greater than windspeed are not going directly downwind. Your original model was more relevant, just your sail model was bad.

That was in reply to the following post #29. My emphasis.

What is the maximum velocity of the cart in the following image?

View attachment 320367

The original configuration that you refer to was in post #20, so lets move back to that and fix it...I will analyze on any angle a jet(s) impinging on a vane. How do you want me to orient\select geometry for the vane such that the craft will outrun the jet(s) in the direction of the jet for some track angle $\theta$?

Assumptions:

1) The vane geometry must be fixed ( it's not changing from a quarter turn to a half turn etc...)
2) The direction of the impinging fluid jet(s) is fixed to the right $\rightarrow^+$
3) The velocity $w$ of the incoming fluid jet(s) relative to the ground is constant magnitude.
4) The frame which we consider the forces is the inertial frame.

If you have a further stipulation that we need to optimize the vane orientation w.r.t. the craft, as a function of cart velocity, I'll work on that after those conditions\assumptions are agreed upon and I complete a fixed vane orientation result.

 

[erobz]

I would agree even with unrealistic "jet wind" concept, asking consider infinity long sail instead.

I can begin with that. So you want a single jet, infinitely long planar sail oriented relative to the cart such that the force is maximized on the cart in the direction of motion?

 

[A.T.]

How do you want me to orient\select geometry for the vane such that the craft will outrun the jet(s) in the direction of the jet for some track angle $\theta$?

You can orient the sail like in the image below:

w : wind relative to the ground
w' : wind relative to the boat
v: boat velocity relative to the ground

2) The direction of the impinging fluid jet(s) is fixed to the right $\rightarrow^+$

This sounds wrong. The direction of fluid inflow onto the sail (w') is not fixed. It changes with v:

w' = w - v

See blue arrows above.

4) The frame which we consider the forces is the inertial frame.

The force by the air on the sail is frame invariant.

 

[erobz]

You can orient the sail like in the image below:

https://www.physicsforums.com/attachments/322853
w : wind relative to the ground
w' : wind relative to the boat
v: boat velocity relative to the groundThis sounds wrong. The direction of fluid inflow onto the sail (w') is not fixed. It changes with v:

w' = w - v

See blue arrows above.The force by the air on the sail is frame invariant.

So it looks like we can't agree...can't say I'm surprised. The wind caries momentum into the sail,and is carried out in the direction the sail deflects it. This relative wind you keep trying to invoke for the analysis doesn't exist in the inertial frame. You are making up momentum that doesn't exist in the inertial frame. I'm doing the analysis in the inertial frame (a frame fixed to the ground)

 

[A.T.]

So it looks like we can't agree...can't say I'm surprised.

Fortunately real world evidence agrees with me.

The wind caries momentum into the sail,and is carried out in the direction the sail deflects it. This relative wind you keep trying to invoke for the analysis doesn't exist in the inertial frame. You are making up momentum that doesn't exist in the inertial frame.

1) We have empirical data on lift/drag ratios, which are all based on the relative wind. Those are way more realistic than the simplifying assumptions about the air deflection that you have to make.

2) The way you yourself keep drawing the flow (as if the vane was fixed) can only refer to the relative wind, The deflection in the ground frame, where both: the air and the vane are moving looks different.

I'm doing the analysis in the inertial frame (a frame fixed to the ground)

You can do that, but keep in mind: The mass flow rate into your moving control volume depends on the wind relative to the control volume. No matter how much you try to ignore it, the varaying relative wind still comes in.

 

[erobz]

2) The way you yourself keep drawing the flow (as if the vane was fixed) can only refer to the relative wind, The deflection in the ground frame, where both: the air and the vane are moving looks different.

How I draw it, or how it looks is not important. What is important is properly accounting for the change in momentum of the flow w.r.t the stationary inertial frame.

You can do that, but keep in mind: The mass flow rate into your moving control volume depends on the wind relative to the control volume. No matter how much you try to ignore it, the varaying relative wind still comes in.

Yes. Thank you.

 

[Gleb1964]

.. This relative wind you keep trying to invoke for the analysis doesn't exist in the inertial frame. You are making up momentum that doesn't exist in the inertial frame. I'm doing the analysis in the inertial frame (a frame fixed to the ground)

Not quite agree. You can consider inertial frame with the boat at any speed. Been sailing a bit, the apparent (relative) wind is a very real thing, especially when you are sitting in the boat.

 

[erobz]

Not quite agree. You can consider inertial frame with the boat at any speed. Been sailing a bit, the apparent (relative) wind is a very real thing, especially when you are sitting in the boat.

You are in a non-inertial frame when you are on the boat accelerating. Let's just talk about how we do Newtonian Mechanics. If you see something is off with the physics I will present, say something. That is what should be the main focus here.

 

[Gleb1964]

At any moment of acceleration it is possible to induce an instant inertial frame to the boat. That inertial frame is good for understand the correct change of the air momentum.
I would even not complicate the task, but just assume the boat moving with constant speed. Make a solve with the constant boat speed. It is always possible to assume that boat has some resistance, keeping it speed constant. Forgot about how boat is getting to the speed. Make it simple.

 

[erobz]

This is the diagram for your suggested infinite planar sail. My goal is to optimize the angle of the sail w.r.t. the cart $\beta$ such that the component of the force $F$ ( shown in purple ) in the direction of motion is maximized. I.e. I'm hoping to maximize $F \cos \gamma$. Seeing the equations I'm developing I don't suspect to have a prayer of getting there in general, but I'll see what I can come up with.

If you don't like it, shoot it down before I go further.

 

[erobz]

At any moment of acceleration it is possible to induce an instant inertial frame to the boat. That inertial frame is good for understand the correct change of the air momentum.
I would even not complicate the task, but just assume the boat moving with constant speed. Make a solve with the constant boat speed. It is always possible to assume that boat has some resistance, keeping it speed constant. Forgot about how boat is getting to the speed. Make it simple.

Unfortunately it can't work like that. I must form some first order differential equation and look at the limit of said equation as $\frac{dv}{dt} \to 0$.