ألبوم الحلول في الجاذبية

يتحرك الصاروخ بتسارع ثابت ، نصف حركة الصاروخ بالنسبة لمحور الحركة الذي يكون نقطة أصله في النقطة التي ينطلق منها الصاروخ واتجاهه نحو الأعلى.

نجد الارتفاع الذي ينفد فيه الوقود بمساعدة دالة المكان كدالة للزمن المناسبة للحركة بتسارع ثابت. 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»t«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»V«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»t«/mi»«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¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»t«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mspace linebreak=¨newline¨»«/mspace»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»t«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»t«/mi»«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»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»20«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»60«/mn»«msup mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«mspace linebreak=¨newline¨»«/mspace»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»900«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»,«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»000«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«/mstyle»«/math»

لذلك ، نفد الوقود على ارتفاع 900 كم فوق نقطة الإطلاق.

لإيجاد تسارع السقوط الحر على ارتفاع 900 كم فوق سطح الأرض، تم وضع جسم كتلته m على هذا الارتفاع.

نرسم مخطّط القوى الذي يؤثر على الجسم:

نكتب معادلة حركة الجسم، ونعبّر عن تسارعه، وهذا التسارع هو تسارع الجاذبية: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»M«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«menclose notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»*«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»M«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»6«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»67«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»974«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»24«/mn»«/msup»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»6«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»38«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»6«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»900«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»3«/mn»«/msup»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»984«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»14«/mn»«/msup»«/mrow»«mrow»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»13«/mn»«/msup»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»7«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»51«/mn»«mfrac mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»m«/mi»«msup»«mi mathvariant=¨bold¨»s«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mfrac»«/math»

فقط قوة الجاذبية تبذل شغل على الصاروخ ، لذلك يتحقق قانون حفظ الطاقة الميكانيكية.

نظرًا لأن الصاروخ ينتقل عبر مسافات كبيرة ، فإنه يتحرك بتسارع جاذبية متغيرة.

لإيجاد أقصى ارتفاع وصل إليه الصاروخ ، نكتب معادلة حفظ الطاقة ونستخدم طاقة وضع الجاذبية.

نشير إلى النقطة التي نفد فيها الوقود بالنقطة A. ونقطة قمة الارتفاع بالنقطة B.

نكتب معادلة حفظ الطاقة الميكانيكية:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨»KA«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨»KB«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mspace linebreak=¨newline¨»«/mspace»«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 mathcolor=¨#0000FF¨»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfenced mathcolor=¨#0000FF¨»«mfrac»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»M«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mrow»«mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/mrow»«/mfrac»«/mfenced»«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 mathcolor=¨#0000FF¨»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfenced mathcolor=¨#0000FF¨»«mfrac»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»M«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mrow»«mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«/mrow»«/mfrac»«/mfenced»«/math»

سرعة الكرة في النقطة B تساوي صفرًا. احسب سرعة الكرة في النقطة A باستخدام الكينيماتيكا:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mrow»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»V«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»t«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»20«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»60«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»6«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»,«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»000«/mn»«mfrac mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»m«/mi»«mi mathvariant=¨bold¨»s«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

نعوّض السرعتين V_A  ،V_B والارتفاع h_A ، ونحصل من معادلة حفظ الطاقة على معادلة بمتغير واحد  h_B ، وسنحل المعادلة ونجد h_B: 

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لذلك ، يبلغ أقصى ارتفاع للصاروخ 4.45 مليون متر فوق سطح الأرض.