Motion in 2 dimension-example from book

[negation]

Homework Statement

A jetliner flies at 960kmh^-1 relative to the air. it's going from Houston to Omaha, 1290km Northwards. At cruising altitude a wind is blowing east wards at 190kmh^-1. In what direction should the plane fly? How long will the trip take?

Homework Equations

None

The Attempt at a Solution

This is the part I am unclear:

x-component: v' cos Θ + V = 0
y-Component: v' sin Θ + 0 = v

I can understand why V is added to v' cos Θ: both are in the same x-direction.
I don't understand the reason behind the y-component.
Also, what is the reason for setting the equation to be zero?

 

[Jilang]

V would need to be negative for the first term to work. I think that's saying you need to fly a bit into the wind to go in a straight line. (i.e. in the opposite direction to the wind).

 

[Pythagorean]

If you have a vector, v', that's not parallel to either the x or y axis, then it has a component in both the x and y directions. The sin and cos decompose the magnitude of that vector into its x contribution and y contribution. The plane wants to fly North, so it should have a positive y value, but the x value should be zero, so you enforce this condition. You don't want the plane drifting to the west or east.

 

[negation]

If you have a vector, v', that's not parallel to either the x or y axis, then it has a component in both the x and y directions. The sin and cos decompose the magnitude of that vector into its x contribution and y contribution. The plane wants to fly North, so it should have a positive y value, but the x value should be zero, so you enforce this condition. You don't want the plane drifting to the west or east.

As regards the x-component, the question I were to ask myself would be "given the gust blowing east wards (horizontal) at 190kmh^-1, at what horizontal velocity must the plane be flying such that x=0kmh^-1", am I right?

 

[Pythagorean]

If by x you mean the x-component of the overall velocity vector of the plane, then yes. I clarify this because often x is the position, and something like v_x would be horizontal velocity.