קישור להדפסת השאלה -דף ראשון.קישור להדפסת השאלה- דף שני.

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כדור B.
כאשר גוף נע בכיוון נגדי לכיוון הציר , מהירותו שלילית. וכאשר הגוף נע בכיוון הציר מהירותו חיובית. בהתאם לכיוון תנועת הכדורים ולסימן המהירות בגרף ניתן לזהות את הכדור המתואר בגרף.
ברגע תחילת התנועה מהירותו של הכדור המתואר בגרף היא שלילית , כיוון ציר התנועה הוא למטה, לכן הגרף מתאים לתיאור התנועה של כדור הנזרק כלפי מעלה. כדור B.
כדי לא לבלבל בין הכדורים , יש לערוך תרשים המכיל את ציר התנועה כלפי מטה ואת שלושת הכדורים ברגע תחילת תנועתם. בהתאם לנתונים בשאלה. לאחר עריכת התרשים יותר קל לענות על השאלה.
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H=100m
כדור B , נע מראש המגדל ועד לקרקע , הוא נע בהשפעת כוח הכובד בלבד . תנועתו היא תנועה בתאוצה קבועה g, מתנועת כדור B ניתן ללמוד על גובה הבניין.
דרך א': נמצא את המיקום הסופי של כדור B ביחס לציר תנועה שראשיתו בנקודת הזריקה וכיוונו כלפי מטה.
ערך המיקום הסופי ביחס לציר תנועה זה שווה לגובה הבניין.
דרך ב': בעזרת ביטוי ריבוע המהירויות ניתן למצוא את ההעתק תנועת כדור B, מרגע זריקתו ועד שפגע בקרקע.
ערך העתק זה שווה לגובה הבניין.
דרך ג': כדור B נע כלפי מעלה עד לנקודת שיא הגובה , ולאחר מכן הוא יורד כלפי מטה בגרף ניתן למצוא את המרחק עובר הכדור בעלייתו ואת המרחק שעובר הכדור בירידתו . המרחק הכולל שעובר הכדור בירידתו Y2 פחות המרחק שעובר הכדור בעלייתו Y1 שווה לגובה הבניין H.
נתאר מרחקים אלו בתרשים:
בגרף מהירות בתלות בזמן השטח התחום בין הפונקציה לציר הזמן שווה למרחק שעבר הגוף בתנועתו. נמצא את המרחקים Y1 ו- Y2 מתוך הגרף.
ערך המיקום הסופי ביחס לציר תנועה זה שווה לגובה הבניין.
דרך ב': בעזרת ביטוי ריבוע המהירויות ניתן למצוא את ההעתק תנועת כדור B, מרגע זריקתו ועד שפגע בקרקע.
ערך העתק זה שווה לגובה הבניין.
דרך ג': כדור B נע כלפי מעלה עד לנקודת שיא הגובה , ולאחר מכן הוא יורד כלפי מטה בגרף ניתן למצוא את המרחק עובר הכדור בעלייתו ואת המרחק שעובר הכדור בירידתו . המרחק הכולל שעובר הכדור בירידתו Y1 פחות המרחק שעובר הכדור בעלייתו Y2 שווה לגובה הבניין H.
בשימוש בפונקציית המקום זמן ובריבוע המהירויות תאוצת הכובד חייב להיות חיובית מכיוון שכיוון הציר הוא כלפי מטה. בנוסף,כדי לבטא נכון את גובה הבניין בתלות במרחקים האנכיים יש "לראות" את התנועה בגרף. ולערוך תרשים ממנו נוח לבטא את גובה הבניין.
DFGDFH TH RTJ
RJ RTJ
דגעי לחע לדגעכ לדחגע כדלחעגד לחיד םע ם'קע
עןע' ןקכע ן'קעכ ן'קעכ ן'וקעכ ן'ועקכ ן
ל'ח קכעדגע יגע כח
ךךוךחיע כע גכעח כח כעיל
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2. תלמידים יודעים שהשטח התחום בגרף מהירות זמן שווה להעתק התנועה, אך זה לא מספיק כדי למצוא את גובה הבניין.
חשוב "לראות " את התנועה בגרף . לערוך תרשים ולבטא את גובה הבניין בתלות במרק העליה ומרחק הירידה.
2. נוח לבחור את ראשית הציר בנקודת הזריקה.
3. תלמידים יודעים שהשטח התחום בגרף מהירות זמן שווה להעתק התנועה. לא די בזה כדי למצוא את גובה הבניין, לכן חשוב לערוך תרשים ממנו קל לכתוב ביטוי המתאר את גובה הבניין .
4. בכל גרף חשוב " לראות " את התנועה בגרף, במקרה זה הגוף עולה ויורד , חלק מהזמן המהירות בגרף שלילית וחלק מהזמן היא חיובית.
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יש להשתמש בפונקציית המקום זמן של כדור A , ובפונקציית המקום זמן של כדור B . כדי למצוא את המרחק בין הכדורים ברגע t=2s.
גודל המהירות ההתחלתית של כדור B נתון בגרף, לכדור A יש מהירות זהה בגודלה ,אך שונה בסימונה.
כדור A נזרק בכיוון הציר וכדור B נזרק נגד כיוון הציר , לכן מהירותו ההתחלתית של כדור A היא V0 , ושל כדור B היא VO-
נכתוב את פונקציית המקום זמן לכל אחד מהכדורים:
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נגדיר פונקציה חדשה
, המתארת את המרחק בין הכדורים בכל רגע:
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נמצא בעזרת פונקציה זו את המרחק בין הכדורים ברגע t=2s:
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דרך נוספת: ניתן למצוא את המיקום של כל אחד מהכדורים ברגע t=2s , ולהחסיר את המיקומים כדי למצוא את המרחק בין הכדורים:
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נכתוב את פונקציית המקום זמן לכל כדור, נתייחס לערך V0 כאל ערך חיובי.
מהירות זריקת כדור A היא V0 , ומהירות זריקת כדור B היא V0-:
דרך א: נגדיר בעזרת פונקציות המקום זמן של הכדורים פונקציה חדשה
, המתארת את המרחק בין כדור A לכדור B:
דרך ב: ניתן למצוא את מיקום כל כדור כעבור שתי שניות מרגע תחילת תנועתו, ולהחסיר בין המיקומים.
הגודל V0 הוא חיובי , בתיאור מהירות זריקת הכדור B נוסף סימן מינוס.
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1. בגרף מהירות בתלות בזמן שיפוע הגרף שווה לתאוצה.
במקרה זה שלושת הגופים נעים בהשפעת כוח הכובד בלבד , לכן תאוצתם זהה וגודלה g.
2.משמעות נקודת החיתוך עם ציר המהירות היא המהירות ההתחלתית, במקרה זה מהירות הזריקה.
לכל אחד מהכדורים יש מהירות התחלתית שונה.
3. משמעות השטח הכלוא בגרף מהירות בתלות בזמן הוא ההעתק.
אם נתייחס להעתק הכולל של כדור B , לשלושת הכדורים יש העתק זהה . העתק זה שווה לגובה הבניין.
משמעות השיפוע בגרף מהירות זמן היא תאוצה. ומשמעות השטח התחום בגרף זה הוא הוא ההעתק. המהירות ההתחלתית שווה למהירות הזריקה , מהירות זו היא הנקודה בה חוצה הפונקציה את ציר המהירות.
1. בגרף מהירות בתלות בזמן שיפוע הגרף שווה לתאוצה.
במקרה זה שלושת הגופים נעים בהשפעת כוח הכובד בלבד , לכן תאוצתם זהה וגודלה g.
2.משמעות נקודת החיתוך עם ציר המהירות היא המהירות ההתחלתית, במקרה זה מהירות הזריקה.
לכל אחד מהכדורים יש מהירות התחלתית שונה.
3. משמעות השטח הכלוא בגרף מהירות בתלות בזמן הוא ההעתק.
אם נתייחס להעתק הכולל של כדור B , לשלושת הכדורים יש העתק זהה . העתק זה שווה לגובה הבניין.
1. כל הגופים הנעים בהשפעת כוח הכבידה בלבד נעים בתאוצה קבועה שגודלה 9.8 מטר לשנייה בריבוע כלפי מטה.
2. הדרך שעבר כדור A שונה מהדרך שעברו כדורים B ו- C. אבל לשלושתם העתיקם זהים. בחישוב העתק התנועה של כדור B מהגרף יש להחסיר את גודל השטח שמעל הגרף בשטח שמתחת לגרף .
2. הדרך שעבר כדור A שונה מהדרך שעברו כדורים B ו- C. אבל לשלושתם העתיקם זהים. בחישוב העתק התנועה של כדור B מהגרף יש להחסיר את גודל השטח שמעל הגרף בשטח שמתחת לגרף .
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כדור A יפגע בקרקע במהירות גדולה יותר מהמהירות שבה יפגע כדור B בקרקע.
כוח החיכוך הוא כוח המפריע לתנועה , אך מסלול תנועת הכדורים שונה, לכן השפעת הכוח שונה .
ללא כוח החיכוך מהירות כדור B ברגע חזרתו לנקודת הזריקה תהיה שווה בגודלה למהירות הזריקה, קל לראות זאת מביטוי ריבוע המהירויות.
כאשר פועל כוח חיכוך מהירות כדור B ברגע חזרתו לנקודת הפגיעה תהיה קטנה ממהירות הזריקה.
אפשר לחשוב על מקרה שקול של שני כדורים הנזרקים מאותו גובה כלפי מטה, כדור B במהירות קטנה וכדור A במהירות גדולה. לכן כדור A יגיע לקרקע במהירות גדולה יותר.
כאשר פועל כוח חיכוך מהירות כדור B ברגע שכדור B מגיע לנקודת הזריקה מהירותו תהיה קטנה יותר.
אפשר לחשוב על מקרה שקול שני שני כדורים הנזרקים מאותו גובה כלפי מטה, כדור B במהירות קטנה וכדור A במהירות גדולה. לכן כדור A יגיע במהירות גדולה יותר.
כאשר יש חיכוך כדור B חוזר לנקודת הזריקה במהירות קטנה יותר ממהירות הזריקה כלפי מעלה.
לכן המקרה דומה לשני כדורים זהים שנזרקו מאותו גובה כלפי מטה, כדור A נזרק במהירות גדולה וכדור B במהירות קטנה, לכן לכדור B תהיה מהירות קטנה יותר ברגע הפגיעה בקרקע.
לאחר הבנת עקרונות הדינמיקה והכרת כוח החיכוך הרבה יותר קל לפתור שאלה זו.
אפשר גם בעזרת שיקולי אנרגיה , בכל מקרה זאת שאלה היגיון לתלמיד שמכיר רק את עקרונות הקינמטיקה.
אפשר גם בעזרת שיקולי אנרגיה , בכל מקרה זאת שאלה היגיון לתלמיד שמכיר רק את עקרונות הקינמטיקה.
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