Question

15£®ÈçͼËùʾ£¬ÔÚÊúÖ±ÃæÄڱ߽çÏßOAµÄÁ½²à´æÔÚ·½ÏòÏà·´µÄÔÈÇ¿µç³¡£¬´Å¸ÐӦǿ¶È´óСÏàͬ£®ÒÑÖªOAÓëÊúÖ±ÏßOC¼Ð½ÇΪ60¡ã£¬BµãΪ¡ÏCOAƽ·ÖÏßÉϵĵãÇÒOA=$\sqrt{3}$L£¬OB=L£®Ò»´øÕýµçµÄÁ£×ÓÖÊÁ¿Îªm£¬µçÁ¿Îªq£¬´ÓOµãÑØOCÒÔ³õËÙ¶Èv₀ÉäÈë´Å³¡£¬Á£×ÓÇ¡ºÃ¾­¹ý±ß½çÉϵÄAµã£®³·È¥´Å³¡¼ÓÉÏÓëOABÔÚͬһÊúÖ±ÃæÄÚµÄÔÈÇ¿µç³¡£¬Á£×ÓÈÔÒÔËÙ¶Èv₀´ÓOµãÑØOCÉäÈëʱ£¬Á£×Ó¾­¹ýBµã£»ÈôËٶȴóС²»±ä£¬¸Ä±äÁ£×Ó´ÓOµãÉäÈëµÄ·½Ïò£¬Á£×ÓÔڵ糡Öо­¹ýAµã£¬ÇÒËٶȴóСΪ$\sqrt{10}$v₀£¬ÒÑÖªÁ£×Ó´ÓOµ½A¶¯ÄܵÄÔö¼ÓÁ¿ÊÇÀë×Ó´ÓOµ½B¶¯ÄÜÔö¼ÓÁ¿µÄ3±¶£®Çó£º
£¨1£©´Å³¡Ç¿¶ÈBµÄ¿ÉÄÜÖµ£»
£¨2£©O¡¢AÁ½µã¼äµÄµçÊƲ
£¨3£©ÔÈÇ¿µç³¡³¡Ç¿µÄ´óСºÍ·½Ïò£®

Answer 1

·ÖÎö £¨1£©Á£×ÓÔÚ·½ÏòÏà·´µÄ´Å³¡ÖÐ×ö˳¡¢ÄæʱÕë·½ÏòµÄÔÈËÙÔ²ÖÜÔ˶¯£¬Óɼ¸ºÎ¹ØϵÈôÐýתÕûÊý´Î£¬Ç¡ÄÜͨ¹ýAµã£¬Óɼ¸ºÎ¹ØϵÕÒ³öOA=n 2Rsin60¡ã£¬´Ó¶øÓÉÂåÂØ×ÈÁ¦ÌṩÏòÐÄÁ¦¾ÍÄÜÇó³ö´Å¸ÐӦǿ¶ÈµÄ¿ÉÄÜÖµ£®
£¨2£©Óɶ¯Äܶ¨Àí¼°ÌâÉèÌõ¼þµÄÄ©ËٶȾÍÄÜÇó³öOAÁ½µãµÄµçÊƲ
£¨3£©ÓÉÌâÉèÌõ¼þ¼°¶¯Äܶ¨Àí£¬µ±ÈëÉä·½Ïò¸Ä±äʱ£¬´Ó¶øʼĩÁ½µãOBµçÊƲÔÙÓɵçÊƲîÓëµç³¡Ç¿¶È¹Øϵ£¬´Ó¶øÇó³öÔÈÇ¿µç³¡µÄ´óС£®

½â´ð ½â£º£¨1£©´øµçÁ£×ÓÔÚÏ°벿´Å³¡ÖÐ×öÒ»´ÎÔÈËÙÔ²ÖÜÔ˶¯¹ì¼£ÈçͼËùʾ£¬
ÿƫתһ´ÎÑØOA·½ÏòÒƶ¯2Rsin60¡ã£¬Èô¾­¹ýn´Îǡͨ¹ýAµã£¬Óɼ¸ºÎ¹Øϵ¿ÉÖª£º
    n£¨2Rsin60¡ã£©=$\sqrt{3}L$
 ÓÉÂåÂ××ÈÁ¦ÌṩÏòÐÄÁ¦£º$q{v}_{0}B=m\frac{{{v}_{0}}^{2}}{R}$
½âµÃ£ºB=$\frac{nm{v}_{0}}{qL}$  £¨n=1¡¢2¡¢3¡­£©
£¨2£©¸ù¾Ý¶¯Äܶ¨Àí£ºqU_OA=$\frac{1}{2}m{{v}_{A}}^{2}-\frac{1}{2}m{{v}_{0}}^{2}$
½âµÃ£ºU_OA=$\frac{9m{{v}_{0}}^{2}}{2q}$
£¨3£©¸ù¾Ý¶¯Äܶ¨Àí£ºqU_OB=$\frac{1}{2}m{{v}_{B}}^{2}-\frac{1}{2}m{{v}_{0}}^{2}$
Òò$\frac{1}{2}m{{v}_{A}}^{2}-\frac{1}{2}m{{v}_{0}}^{2}$=3¡Á£¨$\frac{1}{2}m{{v}_{B}}^{2}-\frac{1}{2}m{{v}_{0}}^{2}$£©
¹ÊU_OA=3U_OB
¸ù¾ÝÔÈÇ¿µç³¡³¡Ç¿ÓëµçÊƵĹØϵ¿ÉÖª£¬½«OA·ÖΪÏàµÈµÄ3¶Î£¬ÈôOD=$\frac{1}{3}OA$=$\frac{\sqrt{3}}{3}L$£¬ÔòD¡¢BµÈµçÊÆ
ÓÉÓàÏÒ¶¨Àí¿ÉÖªDB=OD=$\frac{\sqrt{3}}{3}L$
¹ÊµÈÊÆÃæDB¡ÎOC£¬¹Ê³¡Ç¿·½ÏòˮƽÏòÓÒ
U_OA=E¡Á$\sqrt{3}Lsin60¡ã$
½âµÃ£ºE=$\frac{3m{{v}_{0}}^{2}}{qL}$
´ð£º£¨1£©´Å³¡Ç¿¶ÈBµÄ¿ÉÄÜֵΪ$\frac{nm{v}_{0}}{qL}$£¨n=1¡¢2¡¢3¡­£©£®
£¨2£©O¡¢AÁ½µã¼äµÄµçÊƲîÊÇ$\frac{9m{{v}_{0}}^{2}}{2q}$£®
£¨3£©ÔÈÇ¿µç³¡³¡Ç¿µÄ´óСΪ$\frac{3m{{v}_{0}}^{2}}{qL}$£¬·½ÏòˮƽÏòÓÒ£®

µãÆÀ ±¾ÌâµÄÄѵãÔÚµÚÈýÎÊ£¬±ä»¯Ö®´¦ÓÐÁ½µã£ºÏÈÊǽ«ÔÈÇ¿´Å³¡¸ÄΪÔÈÇ¿µç³¡£¬ÕâÑùÁ£×Óͨ¹ýAµãµÄºÏλÒÆΪ$\sqrt{3}L$£¬ºóÊÇÓָıäÁËÁ£×ÓµÄÈëÉä·½Ïò£¬Í¬ÑùÒÑÖªÁ£×ӵĺÏλÒÆ´óСΪL£¬ÓÉ´Ë¿É֪ĩËٶȵĹØϵ£¬Óɶ¯Äܶ¨Àí¼°µçÊƲîÓ볡ǿ¹Øϵ¾ÍÄÜÇó³öµç³¡Ç¿¶È´óСºÍ·½Ïò£®