A person moving east finds wind blowing south. If he moves with thrice his velocity without changing direction, then he finds the wind blowing in the south-west direction. Find the magnitude of actual speed of the wind if the velocity of the person is V.
Let W be the velocity of wind, and assume coordinate axes according to convention.
For the first part:
The velocity of the guy is Vi and the relative velocity of wind to the person is -aj, where ‘a’ is the winds relative velocity to the person.
So $$-aj=W-Vi$$ $$W=Vi-aj$$
For the second part:
The velocity of the person is 3Vi, and the velocity of wind (relative) is $-\frac{a}{\sqrt 2}i-\frac{a}{\sqrt 2}j$ (since it’s southwest, so it makes an angle of 45). Again, using the relative velocity equation $$W=3Vi-\frac{a}{\sqrt 2}i-\frac{a}{\sqrt 2}j$$ Now this is very most of you have realized that I am wrong, because I when I equate the above two expressions, by result is is very amusing. I have spent hours in this ‘simple’ question but I don’t want to waste anymore time.
Can you please help me with it?
You could examine the south component (or $-aj)$ of the wind and set up the following relationship
$$W_s^2=W^2-V^2=(2V)^2$$
So, you get
$$W=\sqrt{5}V$$
Equivalently, in your setup, you would have
$$W=Vi-aj$$ $$W=3Vi-ai-aj$$
which allows you to get $a=2V$ and in turn
$$W=\sqrt{V^2+(2V)^2}=\sqrt{5}V$$