an airplane travels 200 miles due west then 150 miles 60 degrees north of west What is the resultant displacement

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- Jan 24th 2010, 09:25 AMleinadwerdnadisplacement problem
an airplane travels 200 miles due west then 150 miles 60 degrees north of west What is the resultant displacement

- Jan 24th 2010, 10:27 AM1005
You need to assign a reference for your displacement vectors and then sum them. Most intuitive for me would be to think of the directions as if they were on a map: west being negative x-axis, east being positive x-axis, north being positive y-axis, and south being negative y-axis. From there, you see the first vector is $\displaystyle -200$ and the second vector is $\displaystyle (150 \angle 180^\circ - 60^\circ)$ or $\displaystyle (150 \angle 120^\circ)$

Drawing pictures helps with vector problems such as this one. Do do vector addition, you need to break each vector into rectangular components, and add them.

x: $\displaystyle -200 + 150 \cos (120^\circ)$

y: $\displaystyle 150\sin (120^\circ)$

Your answer then may look like this:

$\displaystyle (-200 + 150 \cos (120^\circ))\hat i \ + (150\sin (120^\circ))\hat j \$

Where $\displaystyle \hat i \$ and $\displaystyle \hat j \$ are unit vectors designating only direction and not magnitude.

If you wanted the answer expressed in phasor form, then do this:

$\displaystyle (\sqrt{(-200 + 150 \cos (120^\circ))^2 + (150\sin (120^\circ))^2}\angle \arctan (\frac{150\sin (120^\circ)}{-200 + 150 \cos (120^\circ)}))$ - Jan 24th 2010, 11:02 AMskeeter
you can use the law of cosines to find the magnitude of the displacement ...

$\displaystyle d = \sqrt{200^2 + 150^2 - 2(200)(150)\cos(120^\circ)}$

then the law of sines to find the direction of the displacement relative to west ...

$\displaystyle \frac{\sin{\theta}}{150} = \frac{\sin(120^\circ)}{d}

$

$\displaystyle \theta = \arcsin\left[\frac{150\sin(120^\circ)}{d}\right]$ - Jan 24th 2010, 02:40 PMleinadwerdna
for some reason the answer key says the displacement is facing north of east i thought it would be north of west

- Jan 24th 2010, 02:42 PMskeeter
- Jan 24th 2010, 06:32 PMleinadwerdna