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3.11-bo'lim : Tegishli tariflar

6. Samolyot havoda 750 metr masofada yerga parallel ravishda 100 m/s tezlikda uchadi va dastlab radar stansiyasidan 2,5 kilometr uzoqlikda joylashgan. Samolyot va radar stantsiyasi orasidagi masofa qanday tezlikda o'zgaradi (a) dastlab va (b) 30 soniyadan keyin u radar stantsiyasidan o'tib ketganmi?

A sketch of the plane and the radar station. The radar station is the point at the lower left and the plane is the point at the upper right, flying to the left as shown by an arrow. The plane is 750 meters above the ground and is 2.5 kilometers horizontally from the radar station. The line drawn from the radar station up to the plane is labeled “Find Rate This Distance Changing”.
a Samolyot va radar stantsiyasi orasidagi masofa dastlab qanday tezlikda o'zgaradi?
Yechim

Ushbu qism uchun biz \(x' = - 100\) qachon \(x = 2500\) ekanligini bilamiz. Bu holda \(x'\) manfiy bo'lishi kerak, chunki \(x\) bu qismda kamayadi. Shuni ham yodda tutingki, biz \(x\) ni metrga aylantirdik, chunki qolgan barcha miqdorlar metrda.

Mana bu qism uchun eskiz.

A sketch of the plane and the radar station for the first part. The radar station is the point at the lower left and the plane is the point at the upper right with an arrow showing that it is flying to the left, towards the station. The plane is 750 m above the ground, the horizontal distance is labeled \(x\) and the line from the station up to the plane is labeled \(z\).
2-qadam

Biz ushbu qismda \(z'\) ni aniqlamoqchimiz, shuning uchun Pifagor teoremasidan foydalanib, \(x\) va \(z\)ni bog'lash uchun quyidagi tenglamani olamiz.

\[{z^2} = {x^2} + {750^2} = {x^2} + 562500\]
3-qadam

Va nihoyat, keling, buni \(t\) bilan farqlaylik va biz uni \(z'\) uchun ham hal qila olamiz, shunda haqiqiy yechim tez va oson topiladi.

\[2z\,z' = 2x\,x'\hspace{0.5in} \Rightarrow \hspace{0.5in}z' = \frac{{x\,x'}}{z}\]
3-qadam

Ushbu muammoni hal qilish uchun biz faqat \(z\) (Pifagor teoremasidan qayta foydalanish) ni aniqlashimiz va keyin yuqoridagi 3-bosqichdagi tenglamaga ulanishimiz kerak.

\[z = \sqrt {{{2500}^2} + {{750}^2}} = \sqrt {6812500} = 250\sqrt {109} = 2610.0766\]

Ushbu qism uchun ikkalasi orasidagi masofaning o'zgarish tezligi,

\[z' = \frac{{\left( {2500} \right)\left( { - 100} \right)}}{{2610.0766}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 95.7826}}\]


b Samolyot radiolokatsion stansiya ustidan oʻtganidan keyin 30 soniyadan keyin samolyot bilan radiolokatsion stansiya orasidagi masofa qanday tezlikda oʻzgaradi?
Yechim

Ushbu qism uchun biz \(x' = 100\) ekanligini bilamiz va bu holatda ijobiy bo'ladi, chunki \(x\) endi ko'payib boradi, chunki biz quyidagi eskizda ko'rishimiz mumkin.

A sketch of the plane and the radar station for the second part. The plane is now the point at the upper left with an arrow showing that it is flying to the left, away from the station, and the radar station is the point at the lower right. The plane is 750 m above the ground, the horizontal distance is labeled \(x\) and the line from the station up to the plane is labeled \(z\).
2-qadam

Oldingi qismda bo'lgani kabi biz \(z'\) ni aniqlamoqchimiz va bizga kerak bo'ladigan tenglama avvalgi qism bilan bir xil, shuning uchun biz uni ham, uning hosilasini ham bu erda qayta yozamiz.

\[\begin{array}{c}{z^2} = {x^2} + {750^2} = {x^2} + 562500\\ 2z\,z' = 2x\,x'\hspace{0.5in} \Rightarrow \hspace{0.5in}z' = \frac{{x\,x'}}{z}\end{array}\]
3-qadam

Ushbu muammoni hal qilish uchun biz faqat \(x\) va \(z\)ni aniqlashimiz kerak. \(x\) uchun biz samolyot tezligini va uning radar stantsiyasidan o'tganidan keyin 30 soniya davomida uchganligini bilamiz. Shunday qilib, \(x\),

\[x = \left( {100} \right)\left( {30} \right) = 3000\]

\(z\) uchun biz faqat Pifagor teoremasini qayta ishlatishimiz kerak.

\[z = \sqrt {{{3000}^2} + {{750}^2}} = \sqrt {9562500} = 750\sqrt {17} = 3092.3292\]

Bu qism uchun ikkalasi orasidagi masofaning o'zgarish tezligi, keyin,

\[z' = \frac{{\left( {3000} \right)\left( {100} \right)}}{{3092.3292}} = \require{bbox} \bbox[2pt,border:1px solid black]{{97.0143}}\]