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You are given two vectors to add. Draw the resultant then calculate its magnitude and direction. A small plane is seen moving at 90m/s east while drifting north at a speed of 30m/s due to high winds. What is its overall velocity

User Bdavidxyz
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1 Answer

15 votes
15 votes

Let two vectors A and B makes an angle θ between them. The sum of the vectors or the magnitude of the resultant vector is given as,


R=\sqrt[]{A^2+B^2+2AB\cos \theta}

And the direction is given as,


\alpha=\tan ^(-1)((\lvert B\rvert\sin \theta)/(\lvert A\rvert+\lvert B\rvert\cos \theta))

Assuming east as positive x direction and north as positive y direction:

Given that,

Velocity of the plane;


v_p=(90\hat{i})\text{ m/s}

Velocity of the wind;


v_w=(30\hat{j})\text{ m/s}

The angle between North and East direction is,


\theta=90\degree

The resultant velocity is given as,


\begin{gathered} v_r=\sqrt[]{v^2_p+v^2_w+2v_pv_w\cos\theta} \\ =\sqrt[]{\lbrack(90)\rbrack^2+\lbrack(30)\rbrack^2+2\lbrack(90)\rbrack\lbrack(30)\rbrack^{}\cos (90\degree)}\text{ m/s} \\ \approx94.87\text{ m/s} \end{gathered}

The direction is given as,


\begin{gathered} \alpha=\tan ^(-1)((\lvert v_w\rvert\sin \theta)/(\lvert v_p\rvert+\lvert v_w\rvert\cos \theta)) \\ =\tan ^(-1)(\frac{\lvert30\hat{j}\rvert\sin (90\degree)}{\lvert90\hat{i}\rvert+\lvert30\hat{j}\rvert\cos (90\degree)}) \\ =\tan ^(-1)((30)/(90)) \\ \approx18.43\degree \end{gathered}

Therefore, the resultant velocity of the plane is 94.87 m/s and is directed 18.43° towards North-East.

You are given two vectors to add. Draw the resultant then calculate its magnitude-example-1
You are given two vectors to add. Draw the resultant then calculate its magnitude-example-2
User Banana Cake
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