Im not sure how to go about answering this question

Question

A jet plane flying from Darwin, Australia has an air speed of 260m/s in a direction 5 degrees south of west. The plane is in the jet stream which is blowing at 35m/s in a direction 20 degrees south of east. What is the ground speed and heading of the jet measured from the west direction?

Answer 1

Here is a sketch:

You are trying to find the resultant velocity.

It makes sense that if the plane was traveling due west at 260 and the wind was blowing due east at 35 that the resultant velocity would be 260 - 35 = 225 m/s but when they are at funny angles it becomes harder.

To do this problem, break each vector into it's components (x and y) so that you can then combine the x's and then the y's to get the components of the resultant. I show the components of the plane's velocity and the wind's velocity in grey.

For the x-direction (use the trig ratio cosine: cos(θ) = adjacent/hypotenuse):

V-plane_x = 260cos(5°) = 259.01 but let's label this as negative since using the Cartesian coordinate system this would be in the negative x-direction) so V-plane_x = -259.01 m/s

V-wind_x = 35cos(20) = 32.89 m/s

For the y-direction (use the trig ratio sine: sin(θ) = opposite/hypotenuse):

V-plane_y = 260sin(5°) = 22.66 but let's label this as negative since using the Cartesian coordinate system this would be in the negative y-direction) so V-plane_y = -22.66 m/s

V-wind_y = 35sin(20) = 11.97 m/s again in the negative y-direction so V-wind_y = -11.97 m/s

Now add the vectors in the x-direction:

V-resultant_x = V-plane_x + V-wind_x = -259.01 + 32.89 = -226.12 m/s

V-resultant_y = V-plane_y + V-wind_y = -22.66 + -11.97 = -34.63 m/s

To put the resultants back together:

To find the magnitude of the resultant use the Pythagorean Theorem: R = √[(V-resultant_x)² + (V-resultant_y)²] = √[(-226.12)² + (-34.63)²] = √52330.2 = 229 m/s

To find the direction, use the trig ratio tangent (because tan(θ) = opposite/adjacent):

θ = tan^-1(-34.63/-226.12) = tan^-1(0.1531) = 8.71°

Based on the fact that both the x-component and y-component of the resultant are negative, this would equate to 8.71° south of west