Calculating Velocity: Wind Velocity Causes an Airplane to Drift

In this problem, somewhat different from the previous example, we know the total velocity ${\mathbf{\text{v}}}_{\text{tot}}$ and that it is the sum of two other velocities, ${\mathbf{\text{v}}}_{\text{w}}$ (the wind) and ${\mathbf{\text{v}}}_{\text{p}}$ (the plane relative to the air mass). The quantity ${\mathbf{\text{v}}}_{\text{p}}$ is known, and we are asked to find ${\mathbf{\text{v}}}_{\text{w}}$. None of the velocities are perpendicular, but it is possible to find their components along a common set of perpendicular axes. If we can find the components of ${\mathbf{\text{v}}}_{\text{w}}$, then we can combine them to solve for its magnitude and direction. As shown in Figure 3.47, we choose a coordinate system with its x-axis due east and its y-axis due north (parallel to ${\mathbf{\text{v}}}_{\text{p}}$). (You may wish to look back at the discussion of the addition of vectors using perpendicular components in Vector Addition and Subtraction: Analytical Methods.)

Because ${\mathbf{\text{v}}}_{\text{tot}}$ is the vector sum of the ${\mathbf{\text{v}}}_{\text{w}}$ and ${\mathbf{\text{v}}}_{\text{p}}$, its x- and y-components are the sums of the x- and y-components of the wind and plane velocities. Note that the plane only has vertical component of velocity so $v_{\mathrm{p}x}=0$ and $v_{\mathrm{p}y}=v_{\text{p}}$. That is,

and

We can use the first of these two equations to find $v_{\text{w}x}$:

Because $v_{\text{tot}}=\text{38}\text{.}\mathrm{0\; m}/\text{s}$ and $\text{cos 110º}=–0.342$ we have

The minus sign indicates motion west which is consistent with the diagram.

Now, to find $v_{\text{w}\text{y}}$ we note that

Here $v_{\text{tot}y}=v_{\text{tot}}\text{sin 110º}$; thus,

This minus sign indicates motion south which is consistent with the diagram.

Now that the perpendicular components of the wind velocity $v_{\text{w}x}$ and $v_{\text{w}y}$ are known, we can find the magnitude and direction of ${\mathbf{\text{v}}}_{\text{w}}$. First, the magnitude is

so that

The direction is:

giving

The wind’s speed and direction are consistent with the significant effect the wind has on the total velocity of the plane, as seen in Figure 3.47. Because the plane is fighting a strong combination of crosswind and head-wind, it ends up with a total velocity significantly less than its velocity relative to the air mass as well as heading in a different direction.