Question

19£®¶¯Á¿Êغ㶨ÂɺͻúеÄÜÊغ㶨ÂÉÊÇÎÒÃǽâ¾öÎïÀíÎÊÌâ³£ÓõÄÀíÂÛÒÀ¾Ý£¬ÇëºÏÀíÀûÓÃÕâÁ½ÎïÀíÀíÂÛÒÀ¾Ý£¬½â¾öÒÔÏÂÎïÀíÇé¾°ÖеľßÌåÎÊÌ⣺Á½ÖÊÁ¿·Ö±ðΪ2MºÍ4MµÄÅüAºÍB£¬¸ß¶ÈÏàͬ£¬·ÅÔڹ⻬ˮƽÃæÉÏ£¬AºÍBµÄÇãбÃ涼Êǹ⻬ÇúÃ棬ÇúÃæ϶ËÓëˮƽÃæÏàÇУ¬ÈçͼËùʾ£¬Ò»ÖÊÁ¿ÎªMµÄÎï¿éλÓÚÅüAµÄÇãбÃæÉÏ£¬¾àˮƽÃæµÄ¸ß¶ÈΪh£®Îï¿é´Ó¾²Ö¹¿ªÊ¼Ï»¬£¬È»ºóÓÖ»¬ÉÏÅüB£®Çó£º
£¨1£©Îï¿éµÚÒ»´ÎÏ»¬µ½×îµÍµãʱ»¬¿éËٶȵĴóС£»
£¨2£©Îï¿éµÚÒ»´ÎÏ»¬µ½×îµÍµãʱÅüAËٶȵĴóС£»
£¨3£©µ±»¬¿éµ½´ïÅüBµÄ×î¸ßµãʱ£¬ÅüBµÄËٶȴóС£»
£¨4£©»¬¿éÄܹ»µ½´ïÅüBµÄ×î´ó¸ß¶ÈH£®

Answer 1

·ÖÎö £¨1¡¢2£©Îï¿é´Ó¿ªÊ¼Ï»¬µ½µÚÒ»´ÎÀ뿪ÅüAʱµÄ¹ý³ÌÖУ¬Îï¿éÓëбÅüA×é³ÉµÄϵͳˮƽ·½Ïò²»ÊÜÍâÁ¦£¬ÏµÍ³Ë®Æ½·½Ïò¶¯Á¿Êغ㣬»úеÄÜÒ²Êغ㣬½áºÏ¶¯Á¿Êغ㶨ÂɺͻúеÄÜÊغ㶨ÂÉÇó³öÎï¿éµÚÒ»´ÎÏ»¬µ½×îµÍµãʱ»¬¿éºÍAµÄËٶȣ»
£¨3£©µ±Îï¿éÔÚÅüBÉÏ´ïµ½×î´ó¸ß¶Èʱ£¬Îï¿éÓëÅüBµÄËÙ¶ÈÏàͬ£¬½áºÏ¶¯Á¿Êغ㶨ÂÉÇó¹²Í¬Ëٶȣ®
£¨4£©ÔÙÓÉ»úеÄÜÊغãÇó³öÎï¿éÔÚÅüBÉÏÄܹ»´ïµ½µÄ×î´ó¸ß¶ÈH£®

½â´ð ½â£º£¨1¡¢2£©ÉèÎï¿éµÚÒ»´ÎÏ»¬µ½×îµÍµãʱʱµÄËÙ¶ÈΪv₁£¬´ËʱÅüAµÄËÙ¶ÈΪv₂£¬¹æ¶¨ÏòÓÒΪÕý·½Ïò£¬ÓÉϵͳˮƽ¶¯Á¿ÊغãµÃ£º
  Mv₁-2Mv₂=0
ÓÉϵͳ»úеÄÜÊغãµÃ£ºMgh=$\frac{1}{2}$Mv₁²+$\frac{1}{2}$•2Mv₂²£®
½âµÃ£ºv₁=$\sqrt{\frac{4}{3}gh}$£¬v₂=$\sqrt{\frac{1}{3}gh}$£®
£¨3¡¢4£©ÉèÎï¿éÔÚÅüBÉÏ´ïµ½×î´ó¸ß¶ÈHʱÁ½ÕßËÙ¶ÈÏàͬ£¬¹²Í¬ËÙ¶ÈΪv£¬¹æ¶¨ÏòÓÒΪÕý·½Ïò£¬ÓÉÎï¿éºÍB×é³ÉµÄϵͳˮƽ¶¯Á¿ÊغãºÍ»úеÄÜÊغãµÃ£º
   Mv₁=£¨M+4M£©v
  $\frac{1}{2}$Mv₁²=$\frac{1}{2}$£¨M+4M£©v²+MgH
½âµÃ£ºv=$\frac{2}{5}$$\sqrt{\frac{1}{3}gh}$£¬H=$\frac{8}{15}$h
´ð£º£¨1£©Îï¿éµÚÒ»´ÎÏ»¬µ½×îµÍµãʱ»¬¿éËٶȵĴóСΪ$\sqrt{\frac{4}{3}gh}$£®
£¨2£©Îï¿éµÚÒ»´ÎÏ»¬µ½×îµÍµãʱÅüAËٶȵĴóСΪ$\sqrt{\frac{1}{3}gh}$£®      
£¨3£©µ±»¬¿éµ½´ïÅüBµÄ×î¸ßµãʱ£¬ÅüBµÄËٶȴóСΪ$\frac{2}{5}$$\sqrt{\frac{1}{3}gh}$£®
£¨4£©»¬¿éÄܹ»µ½´ïÅüBµÄ×î´ó¸ß¶ÈHÊÇ$\frac{8}{15}$h£®

µãÆÀ ±¾Ì⿼²éÁ˶¯Á¿Êغ㶨ÂɺͻúеÄÜÊغ㶨ÂɵÄ×ÛºÏÔËÓ㬹ؼüÑ¡ºÃÑо¿µÄϵͳ£¬Ñ¡ÔñºÏÊʵĹæÂɽøÐÐÇó½â£¬ÖªµÀÎï¿éÉÏÉýµÄ×î´ó¸ß¶ÈʱÓëÅüB¾ßÓÐÏàͬµÄËٶȣ®Îï¿éÀ뿪ÅüBʱÅüBËùÄÜ»ñµÃµÄËٶȴﵽ×î´ó£®