r the thrust of aiplane's engine produces a speed of 350 mph in still air . the plane is aimed in the direction of <6,-3,2> but its velocity with respect to the ground is <320,-140,125> mph .find the wind velocity .explain whether this is a tail wind or head wind .
since the plane is aimed basically SW, but its velocity is basically SE, I'd call it a tail wind, since its southward direction is increased. That's quite a side wind, though, as well.
Does the vertical speed matter when considering this?
To find the wind velocity, we can use the concept of vector addition. The velocity of the airplane with respect to the ground can be considered as the sum of the velocity due to the airplane's thrust and the velocity due to the wind.
Let's denote the wind velocity as <w1, w2, w3> mph.
The velocity due to the airplane's thrust can be calculated by subtracting the wind velocity from the velocity of the airplane in still air:
<350 - w1, 0 - w2, 0 - w3> mph.
The total velocity of the airplane with respect to the ground is given as <320, -140, 125> mph.
So, we have the equation:
<350 - w1, 0 - w2, 0 - w3> + <w1, w2, w3> = <320, -140, 125>.
By combining the components, we get three separate equations:
350 - w1 + w1 = 320,
- w2 + w2 = -140,
- w3 + w3 = 125.
Simplifying each equation, we find:
-w1 = -30,
0 = -140,
0 = 125.
From the second and third equations, we can see that there is no solution for w2 and w3, respectively. This means that there is no wind component in the y and z directions. However, the first equation gives us the wind component in the x direction: w1 = 30 mph.
Since the wind velocity is positive (w1 = 30), we can conclude that this is a tailwind. A tailwind refers to the wind blowing in the same direction as the airplane's motion, which helps increase the airplane's speed.