Question

3£®Ôڹ⻬¾øÔµµÄˮƽÃæÉÏ£¬´æÔÚƽÐÐÓÚˮƽÃæÏòÓÒµÄÔÈÇ¿µç³¡£¬µç³¡Ç¿¶ÈΪE£¬Ë®Æ½ÃæÉÏ·ÅÖÃÁ½¸ö¾²Ö¹£¬ÇÒ¾ù¿É¿´×öÖʵãµÄСÇòAºÍB£¬Á½Ð¡ÇòÖÊÁ¿¾ùΪm£¬AÇò´øµçºÉÁ¿Îª+Q£¬BÇò²»´øµç£¬A¡¢BÁ¬ÏßÓëµç³¡ÏßƽÐУ¬¿ªÊ¼Ê±Á½ÇòÏà¾àL£¬Ôڵ糡Á¦×÷ÓÃÏÂAÇòÓëBÇò·¢Éú¶ÔÐĵ¯ÐÔÅöײ£¬ÉèÅöײ¹ý³ÌÖУ¬A¡¢BÁ½Çò¼äÎÞµçÁ¿×ªÒÆ£®
£¨1£©µÚÒ»´ÎÅöײ½áÊø˲¼äA¡¢BÁ½ÇòµÄËٶȸ÷Ϊ¶à´ó£¿
£¨2£©´Ó¿ªÊ¼µ½¼´½«·¢ÉúµÚ¶þ´ÎÅöµ½Õâ¶Î¹ý³ÌÖе糡Á¦×öÁ˶àÉÙ¹¦£¿
£¨3£©´Ó¿ªÊ¼¼´½«·¢ÉúµÚ¶þ´ÎÅöײÕâ¶Î¹ý³ÌÖУ¬ÈôÒªÇóAÔÚÔ˶¯¹ý³ÌÖжÔ×ÀÃæʼÖÕÎÞѹÁ¦ÇҸպò»À뿪ˮƽ×ÀÃ棨v=0ʱ¿Ì³ýÍ⣩£¬¿ÉÒÔÔÚˮƽÃæÄÚ¼ÓÒ»Óëµç³¡Õý½»µÄ´Å³¡£¬Çëд³ö´Å³¡BÓëʱ¼ätµÄº¯Êý¹Øϵ£®

Answer 1

·ÖÎö £¨1£©´ÓA¿ªÊ¼Ô˶¯µ½ÓëBÇòÅöײ£¬Óɶ¯Äܶ¨Àí»òÅ£¶ÙµÚ¶þ¶¨Âɼ°Ô˶¯Ñ§¹«Ê½¿ÉÇóµÃÅöÇ°AµÄËٶȣ¬·¢Éúµ¯ÐÔÅöײ£¬Óɶ¯Á¿Êغ㶨ÂɺͻúеÄÜÊغ㶨ÂÉ¿ÉÒÔÇó³öÅöºóABÁ½ÇòµÄËٶȣ¬¼ÇµÃ½áÂ۵Ļ°£ºÖÊÁ¿ÏàµÈµÄÁ½¸öÇò·¢Éúµ¯ÐÔÅöײËٶȽ»»»£®
£¨2£©ÓÉÓÚ½»»»Ëٶȣ¬ÔòAÇòµÚÒ»´ÎÅöײºóÓɾ²Ö¹ÔÙ¾­µç³¡¼ÓËÙ£¬¶øBÇò×öÔÈËÙÖ±ÏßÔ˶¯£®µ±AÇòÔÙ´Î×·ÉÏBʱ£¬Á½ÕßµÄλÒÆÏàµÈ£¬´Ó¶øÄÜÇó³ö×·ÉϵÄʱ¼ä¼°×·ÉÏʱAµÄËٶȣ®ÓÉÄÜÁ¿Êغ㶨ÂɾÍÄÜÇó³ö¸Ã¹ý³ÌÖе糡Á¦×öµÄ×ܹ¦£®
£¨3£©ÓÉÌâÉèÌõ¼þ£¬Ð¡ÇòAÔÚÔ˶¯¹ý³ÌÖжÔ×ÀÃæʼÖÕÎÞѹÁ¦ÇҸպò»À뿪ˮƽ×ÀÃ棬ÔòÈκÎʱ¿Ì£¨³ýt=0Ö®Í⣩Êܵ½µÄÂåÂØ×ÈÁ¦ÓëÖØÁ¦Æ½ºâ£¬µ«ÔÚˮƽ·½ÏòÊܵ糡Á¦×öÔȼÓËÙÖ±ÏßÔ˶¯£¬ËٶȾùÔÈÔö´ó£¬ÓÉÂåÂØ×ÈÁ¦¹«Ê½F=BQv£¬ÔòÒªÇóB¼õС£¬F=mg²Å²»±ä£¬¾ßÌåµÄҪд³öƽºâʽ£¬²ÅÄÜд³öB¹ØÓÚtµÄ±í´ïʽ£®

½â´ð ½â£¨1£©AÇòµÄ¼ÓËÙ¶È$a=\frac{QE}{m}$£¬ÅöÇ°AµÄËÙ¶È${v_{A1}}=\sqrt{2aL}=\sqrt{\frac{2QEL}{m}}$£¬ÅöÇ°BµÄËÙ¶Èv_B1=0£®
  ÉèÅöºóA¡¢BÇòËٶȷֱðΪv'_A1¡¢v'_B1£¬Á½Çò·¢ÉúÅöײʱ£¬Óɶ¯Á¿ÊغãºÍÄÜÁ¿Êغ㶨ÂÉÓУº
   mv_A1=mv¡ä_A1+mv¡ä_B1
   $\frac{1}{2}m{{v}^{2}}_{A1}=\frac{1}{2}mv{{¡ä}^{2}}_{A1}+\frac{1}{2}mv{{¡ä}^{2}}_{B1}$
  ËùÒÔAÓëBÅöײºó½»»»Ëٶȣºv'_A1=0£¬${v'_{B1}}={v_{A1}}=\sqrt{\frac{2QEL}{m}}$
£¨2£©ÉèAÇò¿ªÊ¼Ô˶¯Ê±Îª¼ÆʱÁãµã£¬¼´t=0£¬A¡¢BÇò·¢ÉúµÚÒ»´Î¡¢µÚ¶þ´ÎµÄÅöײʱ¿Ì·Ö±ðΪt₁¡¢t₂
  ¶ÔAÇòÓÉÔȱäËٶȹ«Ê½ÓУº${t_1}=\frac{{{v_{A1}}-0}}{a}=\sqrt{\frac{2mL}{QE}}$
  µÚÒ»´ÎÅöºó£¬¾­t₂-t₁ʱ¼äA¡¢BÁ½Çò·¢ÉúµÚ¶þ´ÎÅöײ£¬ÉèÅöǰ˲¼äA¡¢BÁ½ÇòËٶȷֱðΪv_A2ºÍv_B2
ÓÉλÒƹØϵÓУº${v'_{B1}}£¨{t_2}-{t_1}£©=\frac{1}{2}a{£¨{t_2}-{t_1}£©^2}$
  ½âµÃ£ºt₂=3t₁=3$\sqrt{\frac{2mL}{QE}}$
  ${v_{A2}}=a£¨{t_2}-{t_1}£©=2a{t_1}=2{v_{A1}}=2\sqrt{\frac{2QEL}{m}}$
  v_B2=v'_B1=$\sqrt{\frac{2QEL}{m}}$
 Óɹ¦ÄܹØϵ¿ÉµÃ£ºW_µç=$\frac{1}{2}mv_{A2}^2+\frac{1}{2}mv_{B2}^2=5QEL$
[Áí½â£ºÁ½¸ö¹ý³ÌAÇò·¢ÉúµÄλÒÆ·Ö±ðΪx₁¡¢x₂£¬x₁=L£¬ÓÉÔȱä¹æÂÉÍÆÂÛx₂=4L£¬¸ù¾Ýµç³¡Á¦×ö¹¦¹«Ê½ÓÐW=QE£¨x₁+x₂£©=5QEL]
£¨3£©¶ÔAÇòÔÚÊúÖ±·½ÏòÉÏÓÉƽºâÌõ¼þµÃ£º$\left\{\begin{array}{l}{QB{v}_{A}=mg}\\{{v}_{A}=at}\\{a=\frac{QE}{m}}\end{array}\right.$ 
  ´ÓA¿ªÊ¼Ô˶¯µ½·¢ÉúµÚÒ»´ÎÅöײ£º${B_{£¨t£©}}=\frac{mg}{Qat}=\frac{{{m^2}g}}{{{Q^2}Et}}$   £¨$0£¼t¡Ü\sqrt{\frac{2mL}{QE}}$£©
  ´ÓµÚÒ»´ÎÅöײµ½·¢ÉúµÚ¶þ´ÎÅöײ£º${B_{£¨t£©}}=\frac{{{m^2}g}}{\begin{array}{l}{Q^2}E£¨t-\sqrt{\frac{2mL}{QE}}£©\end{array}}$   £¨$\sqrt{\frac{2mL}{QE}}£¼t¡Ü3\sqrt{\frac{2mL}{QE}}$£©
´ð£º£¨1£©µÚÒ»´ÎÅöײ½áÊø˲¼äA¡¢BÁ½ÇòµÄËٶȸ÷Ϊ0¡¢$\sqrt{\frac{2QEL}{m}}$£®
£¨2£©´Ó¿ªÊ¼µ½¼´½«·¢ÉúµÚ¶þ´ÎÅöµ½Õâ¶Î¹ý³ÌÖе糡Á¦×öÁË5QELµÄ¹¦£®
£¨3£©ÈôÒªÇóAÔÚÔ˶¯¹ý³ÌÖжÔ×ÀÃæʼÖÕÎÞѹÁ¦ÇҸպò»À뿪ˮƽ×ÀÃ棨v=0ʱ¿Ì³ýÍ⣩£¬¿ÉÒÔÔÚˮƽÃæÄÚ¼ÓÒ»Óëµç³¡Õý½»µÄ´Å³¡£¬ÄÇô´Å³¡BÓëʱ¼ätµÄº¯Êý¹ØϵʽΪ£º¢Ù´ÓA¿ªÊ¼Ô˶¯µ½·¢ÉúµÚÒ»´ÎÅöײ£º${B_{£¨t£©}}=\frac{mg}{Qat}=\frac{{{m^2}g}}{{{Q^2}Et}}$  £¨$0£¼t¡Ü\sqrt{\frac{2mL}{QE}}$£©£»¢Ú´ÓµÚÒ»´ÎÅöײµ½·¢ÉúµÚ¶þ´ÎÅöײ£º${B_{£¨t£©}}=\frac{{{m^2}g}}{\begin{array}{l}{Q^2}E£¨t-\sqrt{\frac{2mL}{QE}}£©\end{array}}$£¨$\sqrt{\frac{2mL}{QE}}£¼t¡Ü3\sqrt{\frac{2mL}{QE}}$£©£®

µãÆÀ ±¾ÌâÊǵ糡Ïà¹Ø֪ʶÓ붯Á¿Êغ㶨ÂɵÄ×ۺϣ¬ËäÈ»AÇòÊܵ糡Á¦£¬µ«ÅöײµÄÄÚÁ¦Ô¶´óÓÚÄÚÁ¦£¬ÔòÅöײǰºó¶¯Á¿ÈÔÈ»Êغ㣮ÓÉÓÚÁ½ÇòµÄÖÊÁ¿ÏàµÈÔòµ¯ÐÔÅöײºó½»»»Ëٶȣ®ÄÇôAÇòµÚÒ»´ÎÅöºó´ÓËÙ¶ÈΪÁã¼ÌÐø×öÔȼÓËÙÖ±ÏßÔ˶¯£¬Ö±µ½·¢ÉúµÚ¶þ´ÎÅöײ£®ÌâÉè¹ý³ÌÖ»ÊÇ·¢ÉúµÚ¶þ´ÎÅöײ֮ǰµÄÏà¹Ø¹ý³Ì£¬ÓÐÉæ¼°µÚ¶þ´ÎÒÔºóÅöײ£¬µ±È»ÎÊÌâ±äµÃ¼òµ¥Ð©£®