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Q69E

Expert-verifiedFound in: Page 125

Book edition
1st Edition

Author(s)
Paul Peter Urone

Pages
1272 pages

ISBN
9781938168000

**Unreasonable Results A commercial airplane has an air speed of ** ** m/s due east and flies with a strong tailwind. It travels ** ** km in a direction ** ** south of east in ** **h. **

**(a) What was the velocity of the plane relative to the ground? **

**(b) Calculate the magnitude and direction of the tailwind’s velocity. **

**(c) What is unreasonable about both of these velocities? **

**(d) Which premise is unreasonable?**

a. $556\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$m/s.

b. $278\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$And $10.{0}^{\xb0}SE$ .

c. The air speed is very high at m/s, which is more than the speed of sound. The wind speed is also very high at m/s. Hence the problem is unreasonable.

d. Either the distance is larger or the time calculated is smaller.

**Velocity is the rate of change in position of an item in motion as seen from a specific frame of reference and measured by a specific time standard.**

The speed of the plane relative to the ground is

$V=\frac{d}{t}\phantom{\rule{0ex}{0ex}}V=\frac{3000km}{1.5hr}\phantom{\rule{0ex}{0ex}}V=2000\raisebox{1ex}{$km$}\!\left/ \!\raisebox{-1ex}{$hr$}\right.$

Velocity in meter

$V=2000\times \frac{3600}{1000}\phantom{\rule{0ex}{0ex}}V=556\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

The speed of the plane relative to the earth is $556$ m/s.

Hence the velocity can be calculated by the following equation:

${V}_{ae}={V}_{ap}+{V}_{pe}$

The component table will be as below:

| X | Y |

${V}_{ap}$ | -280 | 0 |

${V}_{pe}$ | 554 | -48.5 |

Resultant | 274 | -48.5 |

Here we need to find the x component and the y component.

$Xcomponent\phantom{\rule{0ex}{0ex}}{V}_{pex}={V}_{i}\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}{V}_{pex}=\left(556\right)\mathrm{cos}{5}^{\xb0}\phantom{\rule{0ex}{0ex}}{V}_{pex}=554\phantom{\rule{0ex}{0ex}}$

$Ycomponent\phantom{\rule{0ex}{0ex}}{V}_{pey}={V}_{i}\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}{V}_{pey}=\left(-556\right)\mathrm{sin}{5}^{\xb0}\phantom{\rule{0ex}{0ex}}{V}_{pey}=-48.5$

Hence the resultant vector will be

$\stackrel{\rightharpoonup}{R}=\sqrt{{\left(X\right)}^{2}+{\left(Y\right)}^{2}}\phantom{\rule{0ex}{0ex}}\stackrel{\rightharpoonup}{R}=\sqrt{{\left(274\right)}^{2}+{\left(-48.5\right)}^{2}}\phantom{\rule{0ex}{0ex}}\stackrel{\rightharpoonup}{R}=278\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

The velocity of the air relative to the earth is $278$$\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ .

The direction of the velocity will be:

$\begin{array}{rcl}\mathrm{tan}\theta & =& \frac{Y}{X}\\ \mathrm{tan}\theta & =& \frac{48.5}{274}\\ \theta & =& 10.{0}^{\xb0}\end{array}$

The resultant vector has the direction of $10.{0}^{\xb0}SE$ .

c) The airspeed is very high at $556$m/s, which is more than the speed of sound. The wind speed is also very high at $276$m/s. Hence the problem is unreasonable.

d) The distance may be larger, or the time calculated be smaller.

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