To find the true velocity of wind from relative velocity

• gnits
In summary: So, in summary, Mitch's diagram is not correct, the wind speed is not the same in both situations, and the book's answer of 26.13 knots is obtained by solving for V_{w_y} using the half angle formula.
gnits
Homework Statement
To find the true winde velocity from the relative wind velocity
Relevant Equations
Vr = Va - Vb
Please can I ask for help with the following as to where I'm going wrong.

Book answer is 20 knots and 315 degrees

My solution:

In the below diagram I have sketched the two situations, k is the true speed of the wind.

First question is, is my diagram correct?

The velocity of the wind relaive to the ship ##V_{ws}## is given by:

##V_{ws}=V_w - V_s##

Where ##V_w## is the true velocity of the wind and ##V_s## is the true velocity of the ship.

Let i and j be unit vectors in the directions of east and north respectively
So in the two situations I have:

##k\,sin(22.5)i + k\,cos(22.5)j=V_{w_x}i+V_{w_y}j+20i##

and

##-k\,sin(22.5)i+k\,cos(22.5)j=V_{w_x}i+V_{w_y}j+20j##

Equating coefficients of i and j and solving for ##V_{w_x}## I get:

##V_{w_x}=-10##

and so:

##k=\frac{10}{sin(22.5)}=26.13##

and therefore

##V_{w_y}##=26.13*cos(22.5)=24.13

Which leads to a speed of 26.13 which is not the books answer.

Thanks,
Mitch.

If you add the two equations, your two component equations pop out with very little algebra. Shouldn’t your last equation be

##V_{w_y} = 26.13*cos(22.5) - 10## ?

Still not the book answer. Everything you did down to there looks right to me

Actually, I take that back. That correction DOES give the book answer ... to the proper number of significant figures! It’s annoying because you and I know that in the real world that “0” in the ship’s speed just HAS to be significant. But, if the problem doesn’t explicitly say so ...

How did you get Vwy=26.13*cos(22.5)-10 ?
If I compare the coefficients of j in my two component equations do I not get:
k*cos(22.5)=Vwy AND k*cos(22.5)=Vwy+20
Which leads to Vwy=Vwy+20 which has no solution?

Last edited:
gnits said:
##k\,sin(22.5)i + k\,cos(22.5)j=V_{w_x}i+V_{w_y}j+20i##
and
##-k\,sin(22.5)i+k\,cos(22.5)j=V_{w_x}i+V_{w_y}j+20j##
That appears to be four scalar equations but only three scalar unknowns. Can you spot your error from that?

gnits said:
My solution:

In the below diagram I have sketched the two situations, k is the true speed of the wind.

First question is, is my diagram correct?

View attachment 260294

The velocity of the wind relative to the ship ##V_{ws}## is given by:

##V_{ws}=V_w - V_s##

Where ##V_w## is the true velocity of the wind and ##V_s## is the true velocity of the ship.
...

Thanks,
Mitch.
No, your diagram is not correct.

It's the relative velocity of the wind which is at an angle of ±22.5° . The magnitude of the relative velocity likely is different for the two situations.

Rearrange your equation and make the corresponding sketch.

##\vec {V}_{w}=\vec V_s + \vec V_{ws}##

Last edited:
Thanks all for your help, I see now that my original mistake was in the initial diagram. Indeed the speed is not the same in both cases. Have now solved and agree with book.

gnits said:
Thanks all for your help, I see now that my original mistake was in the initial diagram. Indeed the speed is not the same in both cases. Have now solved and agree with book.
Thanks for letting us know that you solved the problem. Homework Helpers here at PF are volunteers. Our only reward is knowing that you have been helped, so, yes it's good that you indicted that you have a solution.

However, it's customary to post your solution. That will bring the thread to a nice conclusion and allow others to learn from this exercise.

 I rewrote the original equations but with a different variable to represent each of the different speeds of the two realtive winds: ##k_1\,sin(22.5)i + k_1\,cos(22.5)j=V_{w_x}i+V_{w_y}j+20i## ##-k_2\,sin(22.5)i+k_2\,cos(22.5)j=V_{w_x}i+V_{w_y}j+20j## Now there are four equations and four unknowns: ##k_1, k_2, V_{w_x}\,and\,V_{w_y}## Solving for the latter two gives the book's answer. I made use of the half angle formula for tan to derive an exact value of ##tan(22.5^{\circ})\,=\,\sqrt{2}-1##

SammyS

1. What is relative velocity and how is it related to wind velocity?

Relative velocity refers to the velocity of an object or fluid relative to another object or fluid. In the context of wind, it is the velocity of the wind relative to a stationary observer. This means that the observer is not moving with the wind and can measure the wind's velocity without any interference from their own movement.

2. Why is it important to find the true velocity of wind from relative velocity?

It is important to find the true velocity of wind from relative velocity because the relative velocity can be affected by the observer's position and movement. By finding the true velocity, we can accurately measure and predict the wind's speed and direction, which is crucial for various applications such as weather forecasting, aviation, and wind energy production.

3. How is the true velocity of wind calculated from relative velocity?

The true velocity of wind can be calculated by adding the relative velocity to the observer's velocity. This can be done using vector addition, where the two velocities are represented by arrows and the true velocity is the resultant of the two arrows. Alternatively, it can also be calculated using the formula: True Velocity = √(Relative Velocity^2 + Observer's Velocity^2).

4. What factors can affect the accuracy of measuring the true velocity of wind from relative velocity?

There are several factors that can affect the accuracy of measuring the true velocity of wind from relative velocity. These include the accuracy of the instruments used, the observer's position and movement, and the presence of other objects or obstacles that can interfere with the wind's flow. Additionally, wind speed and direction can also vary at different heights, so it is important to take measurements at multiple levels to get a more accurate picture of the wind's velocity.

5. How can we improve the accuracy of measuring the true velocity of wind from relative velocity?

To improve the accuracy of measuring the true velocity of wind from relative velocity, it is important to use high-quality instruments that are regularly calibrated and maintained. The observer should also try to minimize their own movement and position themselves in an open area with minimal obstructions. Taking measurements at multiple heights can also help to improve accuracy, as well as using advanced techniques such as remote sensing methods to gather data from a larger area.

• Introductory Physics Homework Help
Replies
20
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
857
• Introductory Physics Homework Help
Replies
16
Views
2K
• Introductory Physics Homework Help
Replies
6
Views
1K
• Introductory Physics Homework Help
Replies
15
Views
277
• Introductory Physics Homework Help
Replies
3
Views
888
• Introductory Physics Homework Help
Replies
2
Views
682
• Introductory Physics Homework Help
Replies
3
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
804
• Linear and Abstract Algebra
Replies
1
Views
800