حلول التدريبات العملية 3- مسارات رأسية

16. 4.5-

نجد الارتفاع h_A الذي يجب أن يقذف الجسم منه بسرعة V₀ بحيث تكون سرعة الجسم في النقطة C  مساوية «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«/msqrt»«/math»

يتم التعبير عن هذا الارتفاع من معادلة حفظ الطاقة:

                                              «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«/mstyle»«/math»

سرعة الجسم بالنقطة A هي V₀.

ارتفاع النقطة C يساوي ضعف نصف قطر الدوران. 

وبناءً عليه سوف نعبر عن ارتفاع تحرير الجسم اللازم لإكمال حركته. 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mn mathvariant=¨bold¨ mathsize=¨20px¨»0«/mn»«/msub»«mn mathvariant=¨bold¨ mathsize=¨20px¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»R«/mi»«/msqrt»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mn mathvariant=¨bold¨ mathsize=¨20px¨»0«/mn»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«/math»

نضرب المعادلة في 2 ونعبر عن h_A: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»4«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»V«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«msup»«msub»«mi mathvariant=¨bold¨»V«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«/mrow»«/mfrac»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨»V«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«/mrow»«/mfrac»«/menclose»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/mstyle»«/math»