אלבום פתרונות כבידה - שיקולי אנרגיה

נמצא את האנרגיה המינמאלית שיש להשקיע כדי להניע חללית מנקודה A לנקודה O . 

נסמן את מסת החללית ב m3 . מכיוון שמושקעת אנרגיה מנימאלית נתייחס למקרה שבו אין לחללית אנרגיה קינטית בנקודה O, החללית רק מגיעה לנקודה O . נניח שהחללית התחילה לנוע ממנוחה.

האנרגיה שיש להעניק לחללית שווה לעבודת הכוח הלא משמר המניע את החללית בין נקודות אלו. 

נמצא עבודה זו בעזרת ביטוי עבודת הכוח הלא משמר:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»EK«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨normal¨ mathcolor=¨#0000FF¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»EK«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/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»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«msub»«mi mathvariant=¨bold¨»EK«/mi»«mi mathvariant=¨bold¨»O«/mi»«/msub»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«msub»«mi mathvariant=¨bold¨»EK«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mspace linebreak=¨newline¨/»«/math»

החללית נעה בתאוצת כובד משתנה , נשתמש בביטוי האנרגיה הפוטנציאלית כבידתית:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«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¨»U«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»,«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»59«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mrow»«mi mathvariant=¨bold¨»O«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mrow»«mi mathvariant=¨bold¨»O«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»,«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»54«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»6«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨»O«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»54«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»6«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mfrac»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»59«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«/mrow»«/mfenced»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»54«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»6«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/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»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»59«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«/math»

בהתאם לשאלה , יש לבטא את העבודה בתלות ב g , נשתמש ביחס המסות ונבטא את העבודה בתלות ב m1:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»54«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»6«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/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»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»59«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»54«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»81«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»6«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»81«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»59«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»81«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»59«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»54«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mrow»«mn mathvariant=¨bold¨»81«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»6«/mn»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mrow»«mn mathvariant=¨bold¨»81«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»59«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»54«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mrow»«mn mathvariant=¨bold¨»81«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»6«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»4«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»18«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»51«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»05«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»10«/mn»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»98«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msub»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«/math»

יש לבטא את העבודה בתלות בתאוצת הכובד על פני כדור הארץ g . נמצא ביטוי ל g ממשוואת התנועה של גוף הנע בנפילה חופשית על פני כדור הארץ:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/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»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#183;«/mo»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«/mrow»«msup»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8658;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/math»

נציב את ביטוי Gm1 בביטוי העבודה:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»98«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»G«/mi»«mo mathcolor=¨#FF0000¨ mathvariant=¨bold¨»§#183;«/mo»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»m«/mi»«mn mathcolor=¨#FF0000¨ mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»98«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»g«/mi»«mo mathcolor=¨#FF0000¨ mathvariant=¨bold¨»§#183;«/mo»«msup mathcolor=¨#FF0000¨»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»R«/mi»«menclose notation=¨updiagonalstrike¨»«mn mathvariant=¨bold¨»2«/mn»«/menclose»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»R«/mi»«/menclose»«/mfrac»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«msub»«mi mathvariant=¨bold¨»W«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mi mathvariant=¨bold¨»O«/mi»«/mrow»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»98«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mn mathvariant=¨bold¨»3«/mn»«/menclose»«mspace linebreak=¨newline¨»«/mspace»«mspace linebreak=¨newline¨»«/mspace»«/math»

לכן , האנרגיה הנדרשת כדי להניע חללית שמסתה m3 מהנקודה A לנקודה O ,היא : 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨28px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/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¨»98«/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¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«/mrow»«/mstyle»«/math»