30. 1982,3- الحركة بين مسارين
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نصف قطر المدار الأول 24.9 مليون متر.
من التعبير عن الطاقة اللازمة لرفع القمر الاصطناعي ، يتم الحصول على معادلة بمتغيرين R1 و R2.
من التعبير عن النسبة زمني الدورة من القانون الثالث لكبلرنحصل على معادلة أخرى بنفس المجهولين.
يمكنك حل المعادلتين وإيجاد نصف قطر المدار.
מביטוי יחס זמני המחזור מהחוק השלישי של קפלר מתקלת משוואה נוספת עם אותם שני הנעלמים.
ניתן לפתור את המשוואות ולמצוא את רדיוסי המסלול.
الطاقة اللازمة لرفع القمر الاصطناعي من مدار إلى آخر تساوي الفرق في الطاقة الميكانيكية للقمر الاصطناعي في هذين المدارين.
نكتب تعبيرًا عن الطاقة المطلوبة «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«/math»:
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linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»8«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨»=«/mo»«mfenced»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«/menclose»«mspace linebreak=¨newline¨»«/mspace»«/math»
معطى النسبة بين زمني الدورة، في كلا المسارين يتحرك القمر الاصطناعي تحت تأثير الجاذبية فقط، نستخدم قانون كبلر الثالث للتعبير عن النسبة بين نصفي قطر المسارين باستخدام النسبة بين زمني الدورة:
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حصلنا على معادلتين في مجهولين اثنين ، نجد منهما على R1:
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لذلك فإن نصف قطر المدار الأول هو 24.9 مليون متر.
נכתוב ביטוי לאנרגיה הדרושה «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«/math»:
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mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn 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mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»6«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»9«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»6«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»67«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»974«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»24«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»8«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨»=«/mo»«mfenced»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«/menclose»«mspace linebreak=¨newline¨»«/mspace»«/math»
נתון יחס זמני המחזור ,בשני המסלולים הלוויין נע בהשפעת כוח הכבידה בלבד, נשתמש בחוק השלישי של קפלר כדי לבטא באמצעות יחס זמני המחזור את יחס רדיוסי המסלול:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac mathcolor=¨#0000FF¨»«mstyle displaystyle=¨true¨»«msup»«msub»«mi mathvariant=¨bold¨»T«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mstyle»«msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»3«/mn»«/msup»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mstyle displaystyle=¨true¨»«msup»«msub»«mi mathvariant=¨bold¨»T«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mstyle»«msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mn mathvariant=¨bold¨»3«/mn»«/msup»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mfrac mathcolor=¨#0000FF¨»«mstyle displaystyle=¨true¨ mathvariant=¨bold¨»«msup»«msub»«mi»R«/mi»«mn»2«/mn»«/msub»«mn»3«/mn»«/msup»«/mstyle»«mstyle displaystyle=¨true¨ mathvariant=¨bold¨»«msup»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«mn»3«/mn»«/msup»«/mstyle»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»T«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msup»«msub»«mi mathvariant=¨bold¨»T«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup»«mfenced mathcolor=¨#0000FF¨»«mfrac»«mstyle displaystyle=¨true¨»«msub»«mi mathvariant=¨bold¨»T«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mstyle»«mstyle displaystyle=¨true¨»«msub»«mi mathvariant=¨bold¨»T«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mstyle»«/mfrac»«/mfenced»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»8«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»64«/mn»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mfrac mathcolor=¨#0000FF¨»«mstyle displaystyle=¨true¨ mathvariant=¨bold¨»«msub»«mi»R«/mi»«mn»2«/mn»«/msub»«/mstyle»«mstyle displaystyle=¨true¨ mathvariant=¨bold¨»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«/mstyle»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup»«mfenced mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»64«/mn»«/mfenced»«mfrac bevelled=¨true¨ mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»3«/mn»«/mfrac»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»4«/mn»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/menclose»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«/math»
קבלנו שתי משוואות בשני נעלמים, נמצא מהם את R1:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«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¨»M«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»6«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»9«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»6«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»67«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»974«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»24«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»8«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨»=«/mo»«mfenced»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«/menclose»«mspace linebreak=¨newline¨»«/mspace»«/math»
في هذا
السؤال، من الصعب جدًا معرفة الإستراتيجية مسبقًا، فأنت تدرك أنك بحاجة إلى إيجاد R1، وفهم معنى التغيير في الطاقة، وكتابة المعادلة بطريقة واضحة،
وتحصل على معادلة بمجهولين هما R1 و R2. أنت تدرك أنك بحاجة إلى معادلة أخرى بنفس المجهولين، بالنظر إلى النسبة
بين زمني الدورة، من القانون الثالث لكبلر، يمكنك ربط هذه النسبة بنصف قطر المدار،
وهكذا نحصل على المعادلة الثانية.
مع مثل هذه الأسئلة، يوصى بكتابة كل شيء، ومعرفة كيف يمكن المضي قدمًا من كل ما كتبناه.
בשאלות כאלו מומלץ לכתוב הכל , ולראות כיצד מכל מה שכתבנו הכי נכון להתקדם.
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زمن دوران القمر الاصطناعي حول الأرض هو 39107.1 ثانية.
معادلة الحركة
نعبّر عن زمن الدورة، من معادلة حركة القمر الاصطناعي:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#969;«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mspace linebreak=¨newline¨»«/mspace»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»M«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#969;«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»M«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«/mrow»«msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mfenced mathcolor=¨#0000FF¨»«mfrac»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»§#960;«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»T«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«/mfenced»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»T«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mi mathvariant=¨bold¨»§#960;«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»3«/mn»«/msup»«/mrow»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»M«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»T«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mfrac»«mrow»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mi mathvariant=¨bold¨»§#960;«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»3«/mn»«/msup»«/mrow»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»M«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mrow»«/mfrac»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mfrac»«mrow»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mi mathvariant=¨bold¨»§#960;«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold-italic¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»3«/mn»«/msup»«/mrow»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»M«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mrow»«/mfrac»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mfrac»«mrow»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mi mathvariant=¨bold¨»§#960;«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mfenced»«mrow»«mn mathvariant=¨bold¨»24«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»9«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»6«/mn»«/msup»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»3«/mn»«/msup»«/mrow»«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»«/mfrac»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mfrac»«mrow»«mn mathvariant=¨bold¨»6«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»094«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»23«/mn»«/msup»«/mrow»«mrow»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»98«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»14«/mn»«/msup»«/mrow»«/mfrac»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»39«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»,«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»107«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»S«/mi»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«/math»
لذلك ، فإن الزمن الذي يستغرقه القمر الاصطناعي في دورانه حول الكرة الأرضية ، في مداره الأول ، هو 39107 ثانية.
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mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»6«/mn»«/msup»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»3«/mn»«/msup»«/mrow»«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»«/mfrac»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mfrac»«mrow»«mn mathvariant=¨bold¨»6«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»094«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»23«/mn»«/msup»«/mrow»«mrow»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»98«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»14«/mn»«/msup»«/mrow»«/mfrac»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»39«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»,«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»107«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»S«/mi»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«/math»
לכן זמן הקפת הלוויין את כדור הארץ ,במסלול הראשון ,הוא 39,107 שניות.
تعتمد حلول جزء كبير من الأسئلة حول الجاذبية على مبادئ الديناميكا ومعادلة الحركة الدائرية.
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الطاقة المطلوبة تساوي التغير في طاقة القمر الاصطناعي في كلا الوضعين.
الطاقة اللازمة لرفع القمر الاصطناعي ثم إطلاقه في المدار الأول تساوي الفرق بين طاقة القمر الاصطناعي عندما يتحرك في المدار الأول وطاقة القمر الاصطناعي عندما يكون على منصة الإطلاق.
عندما يكون القمر الاصطناعي في منصة الإطلاق ،يكون له فقط طاقة وضع الجاذبية ونشير إلى هذه الطاقة بمقدار U0.
عندما يتحرك القمر الاصطناعي في المدار الأول يكون له طاقة حركية وطاقة وضعية للجاذبية، نستخدم الصيغة لحساب الطاقة الميكانيكية الكلية للقمر الاصطناعي لحساب هذه الطاقة، ونشير إلى هذه الطاقة بـ "E".
يتم حساب فرق الطاقة هذا:
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linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mfrac»«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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«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¨»M«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»6«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»67«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»974«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»24«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mrow»«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»«/mrow»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mrow»«mn mathvariant=¨bold¨»24«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»9«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»6«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»6«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»67«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»974«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»24«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mrow»«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»«/mrow»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»24«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»9«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mn mathvariant=¨bold¨»6«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»98«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»17«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»567«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»7«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»01«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»8«/mn»«/mrow»«/msup»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»98«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»17«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»366«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»7«/mn»«/mrow»«/msup»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»43«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«/msup»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»J«/mi»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«/math»
لذلك ، الطاقة المطلوبة لرفع القمر الصناعي هي «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»43«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«/msup»«/math» جول.
כאשר הלוויין נמצא בכן השיגור יש לו רק אנרגיה פוטנציאלית כבידתית נסמן אנרגיה זו ב U0 .
כאשר הלוויין נע במסלול ההקפה הראשון יש לו אנרגיה קינטית ואנרגיה פוטנציאלית כבידתית , נשתמש בנוסחה לחישוב אנרגיה מכנית כוללת של לוויין לחישוב אנרגיה זו, נסמן את האנרגיה הזו ב 'E .
נחשב הפרש אנרגיה זה:
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linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mfrac»«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»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«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¨»M«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mi mathvariant=¨bold¨»E«/mi»«/msub»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace 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من المهم فهم مفهوم العلاقة بين الطاقة المطلوبة وفرق
الطاقة.
مثل شخص لديه 100 شيكل ويريد شراء منتج بـ 150 شيكل، المبلغ الاضافي المطلوب هو 50 شيكل.
على سبيل المثال، يكون القمر الصناعي في البداية على منصة الإطلاق، وسنشير إلى طاقته في هذه الحالة بواسطة X.
دعنا نشير إلى طاقة القمر الصناعي في المدار الأول بواسطة Y.
يتم حساب الطاقة الإضافية اللازمة لوصول القمر الصناعي بعد الإطلاق إلى المدار الأول وفقًا لـ: Y-X.
2. الفرق بين هاتين الطاقتين يساوي الشغل الذي تبذله قوة خارجية لرفع القمر الاصطناعي بعد الإطلاق إلى مسار المدار الأول.
זה הרעיון של ביטוי העבודה אנרגיה של כוח לא משמר.
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