Vectors In Component Form

**Pandp77** · April 9th 2008, 07:19 AM

Hey Everyone,
If Possible need some help with the following Question. I have attempted (1) and (2) and need some help on (3). Thanks. Any Contribution will be appreciated.

A plane can fly at 350 km/h in still air. The pilot wishes to fly from airport A to airport B where AB= (2000i + 5000j) km

1)
1) There is a wind blowing with velocity ( 30i – 20j) km/h. Find the velocity vector, in the form ai + bj, which the pilot should set so that the plane flies directly from A to B.

My Working Out:

AB = p + w

λ(2000i +5000j) = (ai+bj) + (30i-20j)

2000iλ + 5000jλ = (a+30)i + (b-20)j

a+30 = 2000λ ... (1)
b-20 = 5000λ ... (2)

(2)/(1) =

b-20/a+30 = 2.5

b-20 + 2.5 a + 75

b = 2.5 a + 95

__

a2 + b2 = 3502
a2 + (2.5a + 95 ) 2 = 3502
a2 + 6.25a2 + 475a + 9025 = 122500
7.25 a2 + 475a – 113475 = 0

Using the quadratic equation

a = 96.56 and a = - 162.083
b = 2.5 (96.56 ) + 95 = 336.4
b = 2.5 (- 162.083) + 95 = -310.2075

Therefore p = -310i + 336j km/h

2)
How Long will this flight take?

Velocity = Distance / Time

Time = Distance / Velocity

Time = Square Root of [(-310)^2 + (336)^2]
=457 km

Time = 457 / 350 = 1.306

Time = 1 hour and 18 Minutes

3)
a)How long will it take for the plane to fly from A to B in still air?

Thankyou.

**earboth** · April 9th 2008, 07:34 AM

Originally Posted by Pandp77

...

A plane can fly at 350 km/h in still air. The pilot wishes to fly from airport A to airport B where AB= (2000i + 5000j) km
....

3)
a)How long will it take for the plane to fly from A to B in still air?

1. Calculate the length of $\overrightarrow{AB}$ :

$|\overrightarrow{AB}| = \sqrt{2000^2+50002}^=\sqrt{29,000,000}\approx 5,385.2\ km$

2. $time = \frac{distance}{speed}$

$t = \frac{5385.2\ km}{350\ \frac{km}{h}} \approx 15.3863 \ h \approx 15h; 23min;11 s$

**Pandp77** · April 9th 2008, 08:29 AM

Thanks for that earmoth.

But could anyone please see if my working out for question 1 is correct? and explain to me how to do number 3.

Please do help me as this is due today.

Thankyou.

**iamapineapple** · March 22nd 2013, 03:03 AM

Do you still want help? :P

**Soroban** · March 22nd 2013, 06:21 AM

Hello, Pandp77!

$\text{A plane can fly at 350 km/h in still air. \;The pilot wishes to fly}$
$\text{ from airport }A\text{ to airport }B\text{, where }\overrightarrow{AB}\,=\, (2000i + 5000j)$

$\text{(1) There is a wind blowing with velocity }(30i – 20j)\text{ km/hr.}$
. . . $\text{Find the velocity vector, in the form }ai + bj\text{, which the pilot}$
. . . $\text{should set so that the plane flies directly from }A\text{ to }B.$

Code:

$\text{We have: }\:\overrightarrow{AB} \:=\:\langle 2000,5000\rangle,\;\;\overrightarrow{CB} \:=\:\langle 30,\text{-}20\rangle$

$\text{And: }\:\overrightarrow{AC} + \overrightarrow{CB} \:=\:\overrightarrow{AB}$

. $\overrightarrow{AC} + \langle 30,\text{-}20\rangle \:=\:\langle2000,5000\rangle$

. . . . . . . . $\overrightarrow{AC} \:=\:\langle 1970,5020\rangle$

. . . . . . . . $\overrightarrow{AC} \:=\:1970i + 5020j$

$\text{(2) How long will this flight take?}$

$\frac{\text{Distance}}{\text{Speed}} \:=\:\frac{\sqrt{1970^2+5020^2}}{350} \:=\:\frac{\sqrt{29,\!081,\!300}}{350}$

. . . . . . . $=\:\frac{5392.70804}{350} \:=\:15.40773726$

$\text{Time} \;\approx\;15\text{ hours, }24.5\text{ minutes}$

$\text{(3) How long will it take for the plane to fly from }A\text{ to }B\text{ in still air?}$

$\left|\overrightarrow{AB}\right| \:=\:\sqrt{2000^2+5000^2} \:=\:\sqrt{29,\!000}$

$\frac{\text{Distance}}{\text{Speed}} \:=\:\sqrt{29,\!000}}{350} \:=\:15.38618516$

$\text{Time}\;\approx\;15\text{ hours, }23.2\text{ minutes}$

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