Question

18£®ÈçͼËùʾ£¬ÓÐÒ»¹â»¬¹ìµÀABC£¬ABΪÊúֱƽÃæÄڰ뾶ΪRµÄËÄ·ÖÖ®Ò»Ô²»¡¹ìµÀ£¬BC²¿·ÖΪ×ã¹»³¤µÄˮƽ¹ìµÀ£®Ò»¸öÖÊÁ¿Îªm₁µÄСÎïÌå×ÔA´¦Óɾ²Ö¹ÊͷźóÑØÔ²»¡¹ìµÀAB»¬Ï£¬ÓëÔÚˮƽ¹ìµÀBCÉÏÖÊÁ¿Îªm₂µÄ¾²Ö¹ÎïÌåÏàÅö£®

£¨1£©Èç¹ûm₂ÓëˮƽÇᵯ»ÉÏàÁ¬£¬µ¯»ÉµÄÁíÒ»¶ËÁ¬Ôڹ̶¨×°ÖÃPÉÏ£®m₁»¬µ½Ë®Æ½¹ìµÀºóÓëm₂·¢ÉúÅöײµ«²»Õ³Á¬£¬Åöײºóm₁Óëm₂-Æ𽫵¯»ÉѹËõºó±»µ¯»Ø£¬m₁Óëm₂ÖØзֿª£®Èôµ¯»ÉѹËõºÍÉ쳤¹ý³ÌÖÐÎÞ»úеÄÜËðʧ£¬ÇÒm₁=m₂£¬Çóm₁·´µ¯ºóÄÜ´ïµ½µÄ×î´ó¸ß¶È£»
£¨2£©Èç¹ûÈ¥µôÓëm₂ÏàÁ¬µÄµ¯»É¼°¹Ì¶¨×°ÖÃP£¬m₁ÈÔ´ÓA´¦Óɾ²Ö¹ÊÍ·Å£®Èôm₁=$\frac{1}{2}$m₂£¬m₁Óëm₂µÄÅöײ¹ý³ÌÖÐÎÞ»úеÄÜËðʧ£¬ÇóÅöײºóm₁ÄÜ´ïµ½µÄ×î´ó¸ß¶È£»
£¨3£©ÔÚÂú×ãµÚ£¨2£©ÎʵÄÌõ¼þÏ£¬Èôm₁Óëm₂µÄÅöײ¹ý³ÌÖÐÎÞ»úеÄÜËðʧ£¬ÒªÊ¹m₁Óëm₂Ö»ÄÜ·¢ ÉúÒ»´ÎÅöײ£¬Çóm₂Óëm₁µÄ±ÈÖµ·¶Î§£®

Answer 1

·ÖÎö £¨1£©£¨2£©m₁´ÓA»¬µ½B»úеÄÜÊغ㣬ºÍm₂·¢ÉúÅöײʱ¶¯Á¿Êغ㣬¾Ý´Ë¿ÉÇó³öÅöײʱµÄ¹²Í¬Ëٶȣ¬µ±µ¯»É»Ö¸´µ½×ÔÈ»³¤¶Èʱm₁Óëm₂ÖØзֿª£¬´Ëʱm₁Óëm₂µÄËÙ¶ÈÓë´Ë¹²Í¬ËÙ¶ÈÏàµÈ£¬È»ºó¸ù¾Ý»úеÄÜÊغ㼴¿ÉÇó³öm₁ÄÜ´ïµ½µÄ×î´ó¸ß¶È£®
£¨3£©a£®m₁Óëm₂·¢ÉúÅöײʱ¶¯Á¿Êغãͬʱ»úеÄÜÊغ㣬¾Ý´ËÇó³öm₁ÅöºóËٶȣ¬È»ºó¸ù¾Ý»úеÄÜÊغ㼴¿ÉÇó³öm₁ÄÜ´ïµ½µÄ×î´ó¸ß¶È£®
b£®¸ù¾Ým₁Óëm₂·¢ÉúÅöײʱ¶¯Á¿Êغ㡢»úеÄÜÊغãÇó³öËûÃǵÄÅöºóËٶȣ¬×¢ÒâµÚÒ»´ÎÅöºóm₁µÄËٶȴóÓÚm₂µÄËٶȣ¬µÚ¶þ´Îm₁µÄËÙ¶ÈСÓÚm₂µÄËٶȣ¬¾Ý´Ë¿ÉÕýÈ·½â´ð£®

½â´ð ½â£º£¨1£©m₁´ÓA»¬µ½BÖØÁ¦ÊÆÄÜת»¯Îª¶¯ÄÜ£¬m₁µÄËٶȴﵽv₁
${m}_{1}gR=\frac{1}{2}{m}_{1}{v}_{1}^{2}$                  ¢Ù
m₁Óëm₂·¢ÉúÅöײʱµ¯»É´¦ÓÚ×ÔȻ״̬£¬ÏµÍ³¶¯Á¿Êغ㣬ÅöײºóÒÔ¹²Í¬ËÙ¶Èv_¹²ÏòÓÒÔ˶¯£®
m₁v₁+m₂v₂=£¨m₁+m₂£©v_¹² ¢Ú
ÁªÁ¢¢Ù¢Ú½âµÃ£º${v}_{¹²}=\frac{{v}_{1}}{2}=\frac{\sqrt{2gR}}{2}$         
m₁Óëm₂Ò»Æ𽫵¯»ÉѹËõºóÓÖ±»µ¯»Ø£¬µ±µ¯»É»Ö¸´µ½×ÔÈ»³¤¶Èʱm₁Óëm₂ÖØзֿª£¬´Ëʱm₁Óëm₂µÄËٶȶ¼Îªv_¹²£¬m₁ÒÔv_¹²Îª³õËٶȻ¬ÉÏÔ²»¡¹ìµÀ£¬Éèm₁ÄÜ´ïµ½µÄ×î´ó¸ß¶ÈÊÇh
$\frac{1}{2}{m}_{1}{v}_{¹²}^{2}={m}_{1}gh$
½âµÃ $h=\frac{1}{4}R$  
¹Êm₁·´µ¯ºóÄÜ´ïµ½µÄ×î´ó¸ß¶ÈΪ£º$h=\frac{1}{4}R$£®
£¨2£©³·È¥µ¯»É¼°¹Ì¶¨×°Öúó£®
a£®m₁Óëm₂·¢ÉúÅöײʱϵͳ¶¯Á¿Êغ㣬ÇÒûÓлúеÄÜËðʧ£®ÉèÏòÓÒΪÕý·½Ïò£¬ÓÐ
${m}_{1}{v}_{1}={m}_{1}{v}_{1}^{¡ä}+{m}_{2}{v}_{2}^{¡ä}$          ¢Û
$\frac{1}{2}{m}_{1}{v}_{1}^{2}=\frac{1}{2}{m}_{1}£¨{v}_{1}^{¡ä}£©^{2}$$+\frac{1}{2}{m}_{2}£¨{v}_{2}^{¡ä}£©^{2}$          ¢Ü
´úÈë${m}_{1}=\frac{1}{2}{m}_{2}$£¬ÁªÏµ¢Û¢Ü¿ÉµÃ£º${v}_{1}^{¡ä}=-\frac{1}{3}\sqrt{2gR}$£¬¸ººÅ±íʾm₁Ïò×óÔ˶¯
´Ëºóm₁³åÉÏÔ²»¡¹ìµÀ£¬Éèm₁ÄÜ´ïµ½µÄ×î´ó¸ß¶ÈÊÇh¡ä£¬ÓУº
$\frac{1}{2}{m}_{1}£¨{v}_{1}^{¡ä}£©^{2}=mg{h}^{¡ä}$
½«${v}_{1}^{¡ä}$´øÈëÉÏʽ£¬¿ÉµÃ£º${h}^{¡ä}=\frac{1}{9}R$        
¹ÊÅöײºóm₁ÄÜ´ïµ½µÄ×î´ó¸ß¶ÈΪ£º${h}^{¡ä}=\frac{1}{9}R$£®     
b£®m₁»¬µ½Ë®Æ½¹ìµÀÒÔËÙ¶Èv₁Óë¾²Ö¹µÄm₂·¢ÉúµÚÒ»´ÎÅöײ£¬ÉèÏòÓÒΪÕý·½Ïò£¬ÓÐ
${m}_{1}{v}_{1}={m}_{1}{v}_{1}^{¡ä}+{m}_{2}{v}_{2}^{¡ä}$       
$\frac{1}{2}{m}_{1}{v}_{1}^{2}=\frac{1}{2}{m}_{1}£¨{v}_{1}^{¡ä}£©^{2}$$+\frac{1}{2}{m}_{2}£¨{v}_{2}^{¡ä}£©^{2}$       
½âµÃ£º${v}_{1}^{¡ä}=\frac{£¨{m}_{1}-{m}_{2}£©{v}_{1}}{{m}_{1}+{m}_{2}}$£¬${v}_{2}^{¡ä}=\frac{2{m}_{1}{v}_{1}}{{m}_{1}+{m}_{2}}$
ÒªÄÜ·¢ÉúµÚ¶þ´ÎÅöײµÄÌõ¼þÊÇ${v}_{1}^{¡ä}£¼0$£¬¼´m₁£¼m₂£»ÇÒ|${v}_{1}^{¡ä}$|£¾${v}_{2}^{¡ä}$£¬¼´|m₁-m₂|£¾2m₁£¬¿ÉµÃ
m₂£¾3m₁ ¢Ý
m₁´ÓÔ²»¡¹ìµÀÉÏ»¬Ï£¬ÒÔ´óСΪ|${v}_{1}^{¡ä}$|=$\frac{{m}_{2}-{m}_{1}}{{m}_{1}+{m}_{2}}{v}_{1}$µÄËÙ¶ÈÓëËÙ¶ÈΪ${v}_{2}^{¡ä}=\frac{2{m}_{1}{v}_{1}}{{m}_{1}+{m}_{2}}$µÄm₂·¢ÉúµÚ¶þ´ÎÅöײ£¬ÓУº
${m}_{1}|{v}_{1}^{¡ä}|+{m}_{2}{v}_{2}^{¡ä}={m}_{1}{v}_{1}^{¡å}$$+{m}_{2}{v}_{2}^{¡å}$
$\frac{1}{2}{m}_{1}£¨{v}_{1}^{¡ä}£©^{2}+\frac{1}{2}{m}_{2}£¨{v}_{2}^{¡ä}£©^{2}$=$\frac{1}{2}{m}_{1}£¨{v}_{1}^{¡å}£©^{2}+\frac{1}{2}{m}_{2}£¨{v}_{2}^{¡å}£©^{2}$
µÚ¶þ´ÎÅöºóm₁ºÍ m₂µÄËÙ¶È
${v}_{1}^{¡å}=\frac{4{m}_{1}{m}_{2}-{£¨m}_{2}-{m}_{1}£©^{2}}{£¨{m}_{1}+{m}_{2}£©^{2}}{v}_{1}$     ¢Þ
${v}_{2}^{¡å}=\frac{4{m}_{1}£¨{m}_{2}-{m}_{1}£©}{£¨{m}_{1}+{m}_{2}£©^{2}}{v}_{1}$       ¢ß
²»·¢ÉúµÚÈý´ÎÅöײµÄÌõ¼þΪ£º|${v}_{1}^{¡å}$|¡Ü${v}_{2}^{¡å}$
Ôò£º-4m₁£¨m₂-m₁£©¡Ü$4{m}_{1}{m}_{2}-£¨{m}_{2}-{m}_{1}£©^{2}$¡Ü4m₁£¨m₂-m₁£©
½â²»µÈʽ$-4{m}_{1}£¨{m}_{2}-{m}_{1}£©¡Ü4{m}_{1}{m}_{2}-{£¨m}_{2}-{m}_{1}£©^{2}$
µÃ£º$£¨5-2\sqrt{5}£©{m}_{1}¡Ü{m}_{2}¡Ü£¨5+2\sqrt{5}£©{m}_{1}$            ¢à
½â²»µÈʽ£º$4{m}_{1}{m}_{2}-£¨{m}_{2}-{m}_{1}£©^{2}¡Ü4{m}_{1}£¨{m}_{2}-{m}_{1}£©$
µÃ    m₂¡Ý3m₁  »òm₂¡Ü-m₁  ¢á£¨9£©
×ۺϢݡ¢¢à¡¢¢á£¬m₁Óëm₂Ö»ÄÜ·¢ÉúÁ½´ÎÅöײµÄÌõ¼þΪ£º$3£¼\frac{{m}_{2}}{{m}_{1}}¡Ü5+2\sqrt{5}$
¹ÊҪʹm₁Óëm₂Ö»ÄÜ·¢ÉúÁ½´ÎÅöײ£¬m₂Óë m₁µÄ±ÈÖµ·¶Î§Îª£º$3£¼\frac{{m}_{2}}{{m}_{1}}¡Ü5+2\sqrt{5}$£®
´ð£º£¨1£©m₁·´µ¯ºóÄÜ´ïµ½µÄ×î´ó¸ß¶ÈÊÇ$\frac{1}{4}R$£»£¨2£©Åöײºóm₁ÄÜ´ïµ½µÄ×î´ó¸ß¶ÈÊÇ$\frac{1}{9}R$£»£¨3£©m₂Óëm₁µÄ±ÈÖµ·¶Î§ÊÇ$3£¼\frac{{m}_{2}}{{m}_{1}}¡Ü5+2\sqrt{5}$£®

µãÆÀ ±¾Ì⿼²éÁ˶¯Á¿ºÍÄÜÁ¿ÎÊÌ⣬ÓÐÒ»¶¨µÄÄѶȣ¬ÄѵãÔÚÓÚÊýѧÔËË㣬Òò´ËƽʱѵÁ·ÖÐҪעÒâÊýѧ֪ʶÔÚÎïÀíÖеÄÓ¦Óã®