Question

An airplane is flying at a speed of 350 mph at an angle of 100 degrees. A wind is blowing at 50 mph at an angle of 40 degrees. Find the resulting speed and direction.

Answer 1

The resulting speed and direction of the airplane are approximately 377.492 miles per hour and 93.413°, respectively.

How to apply vectors to find absolute velocity of the airplane

In this question we must apply vectors in rectangular form to determine the absolute velocity of an airplane. which is the sum of velocity at still air ($\vec v_{A/W}$) and wind velocity ($\vec v_{W}$), both in miles per hour. If we know that $v_{W} = 50\,\frac{mi}{h}$, $\theta_{W} = 40^{\circ}$, $v_{A/W} = 350\,\frac{mi}{h}$ and $\theta_{A/W} = 100^{\circ}$, then resulting velocity is:

$\vec v_{A} = 50\cdot (\cos 40^{\circ}, \sin 40^{\circ}) + 350\cdot (\cos 100^{\circ}, \sin 100^{\circ})$

$\vec v_{A} = (-22.474, 376.822)$

Whose magnitude ($v_{A}$), in miles per hour, and direction ($\theta$), in degrees, are, respectively:

$v_{A} = \sqrt{(-22.474)^{2}+376.822^{2}}$

$v_{A} \approx 377.492\,\frac{mi}{h}$

$\theta_{A} = \tan^{-1} \left(\frac{376.822}{-22.474} \right)$

$\theta_{A} \approx 93.413^{\circ}$

The resulting speed and direction of the airplane are approximately 377.492 miles per hour and 93.413°, respectively. $\blacksquare$

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