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The airspeed of an airplane is 500 mph and its heading is 60^\circ. A wind is blowing from the...

Question:

The airspeed of an airplane is 500 mph and its heading is {eq}60^\circ. {/eq} A wind is blowing from the south at 40 mph. Find the plane's ground speed and its course.

Ground Speed and Course.

In aviation, the directions are respect to the y-axis, therefore, when working with vectors, it works differently than when working in other scenarios. Example, for the x-axis we work with the sine of the angle and for the y-axis we work with the cosine of the angle. The decomposition of vectors allows us to find the coordinates of the resulting vector, which would be the ground speed.

Answer and Explanation:

Data:

{eq}A=500\:mph \quad \theta=60^{\circ} \\ B=40\:mph \quad \alpha=180^{\circ} {/eq}


We need to find the components of the vectors.

Components of Vector A:

{eq}\begin{align*} A_x&=A\sin\theta \\ A_x&=500\:mph \cdot \sin 60^{\circ} \\ A_x&=433.01\:mph \\ \\ A_y&=A\cos\theta \\ A_y&=500\:mph \cdot \cos 60^{\circ} \\ A_y&=250\:mph \end{align*} {/eq}


Components of Vector B:

{eq}\begin{align*} B_x&=B\sin\alpha \\ B_x&=40\:mph \cdot \sin 180^{\circ} \\ B_x&=0\:mph \\ \\ B_y&=B\cos\alpha \\ B_y&=40\:mph \cdot \cos 180^{\circ} \\ B_y&=-40\:mph \end{align*} {/eq}


The components of the resultant vector are:

{eq}\begin{align*} R_x&=A_x+B_x \\ R_x&=433.01\:mph+0\:mph \\ R_x&=433.01\:mph \\ \\ R_y&=A_y+B_y \\ R_y&=250\:mph-40\:mph \\ R_y&=210\:mph \end{align*} {/eq}


The magnitude of the resultant vector is:

{eq}\begin{align*} R&=\sqrt{R_x^2+R_y^2} \\ R&=\sqrt{\left(433.01\:mph\right)^2+\left(210\:mph\right)^2} \\ R&=481.24\:mph \end{align*} {/eq}


The direction of the resultant vector is:

{eq}\begin{align*} \beta&=\tan^{-1}\left(\frac{R_x}{R_y}\right) \\ \beta&=\tan^{-1}\left(\frac{433.01\:mph}{210\:mph}\right) \\ \beta&=64.12^{\circ} \end{align*} {/eq}


Answer:

The plane's ground speed is 481.24 mph and its course is N64.12E


Learn more about this topic:

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Vectors: Definition, Types & Examples

from Common Entrance Test (CET): Study Guide & Syllabus

Chapter 57 / Lesson 3
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