Шар массой m1 = 3 кг, движущийся со скоростью V, налетает напокоящийся шар и после абсолютно упругого столкновения отскакивает от негопод углом 90 к первоначальному направлению со скоростью V/2. Определитемассу m2 второго шара. Поверхности шаров гладкие.
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Ответил mappku
0
используем законы сохранения импульса и энергии
индексы 1 и 2 первый и второй шар
без штриха до взаимодействия(удара), а со штрихом после
![left { {{overrightarrow{p_1}=overrightarrow{p_1}'+overrightarrow{p_2}'} atop {E_1=E_1'+E_2'}} right. left { {{m_1cdotoverrightarrow{v_1}=m_1cdotoverrightarrow{v_1}'+m_2cdotoverrightarrow{v_2}'} atop {m_1cdot v_1^2=m_1cdot v_1'^2+m_2cdot v_2'^2}} atop {overrightarrow{v_1}cdotoverrightarrow{v_2}=0;} right. \
left|overrightarrow{p_1}right|=sqrt{p_1^2+p_2^2};\](https://tex.z-dn.net/?f=+left+%7B+%7B%7Boverrightarrow%7Bp_1%7D%3Doverrightarrow%7Bp_1%7D%27%2Boverrightarrow%7Bp_2%7D%27%7D+atop+%7BE_1%3DE_1%27%2BE_2%27%7D%7D+right.+++++left+%7B+%7B%7Bm_1cdotoverrightarrow%7Bv_1%7D%3Dm_1cdotoverrightarrow%7Bv_1%7D%27%2Bm_2cdotoverrightarrow%7Bv_2%7D%27%7D+atop+%7Bm_1cdot+v_1%5E2%3Dm_1cdot+v_1%27%5E2%2Bm_2cdot+v_2%27%5E2%7D%7D+atop+%7Boverrightarrow%7Bv_1%7Dcdotoverrightarrow%7Bv_2%7D%3D0%3B%7D+right.+%5C%0Aleft%7Coverrightarrow%7Bp_1%7Dright%7C%3Dsqrt%7Bp_1%5E2%2Bp_2%5E2%7D%3B%5C "left { {{overrightarrow{p_1}=overrightarrow{p_1}'+overrightarrow{p_2}'} atop {E_1=E_1'+E_2'}} right. left { {{m_1cdotoverrightarrow{v_1}=m_1cdotoverrightarrow{v_1}'+m_2cdotoverrightarrow{v_2}'} atop {m_1cdot v_1^2=m_1cdot v_1'^2+m_2cdot v_2'^2}} atop {overrightarrow{v_1}cdotoverrightarrow{v_2}=0;} right. \
left|overrightarrow{p_1}right|=sqrt{p_1^2+p_2^2};\")
![left { {{overrightarrow{p_1}-overrightarrow{p_2}'=overrightarrow{p_1}'} atop {E_1=E_1'+E_2'}} atop {overrightarrow{p_1}cdotoverrightarrow{p_2}=0} right. left { {{m_1cdotoverrightarrow{v_1}-m_2cdotoverrightarrow{v_2}'=m_1cdotoverrightarrow{v_1}'} atop {m_1cdot v_1^2=m_1cdot v_1'^2+m_2cdot v_2'^2}} atop {m_1cdotoverrightarrow{v_1}cdot m_2cdotoverrightarrow{v_2}=0;} right. \](https://tex.z-dn.net/?f=+left+%7B+%7B%7Boverrightarrow%7Bp_1%7D-overrightarrow%7Bp_2%7D%27%3Doverrightarrow%7Bp_1%7D%27%7D+atop+%7BE_1%3DE_1%27%2BE_2%27%7D%7D+atop+%7Boverrightarrow%7Bp_1%7Dcdotoverrightarrow%7Bp_2%7D%3D0%7D+right.++++left+%7B+%7B%7Bm_1cdotoverrightarrow%7Bv_1%7D-m_2cdotoverrightarrow%7Bv_2%7D%27%3Dm_1cdotoverrightarrow%7Bv_1%7D%27%7D+atop+%7Bm_1cdot+v_1%5E2%3Dm_1cdot+v_1%27%5E2%2Bm_2cdot+v_2%27%5E2%7D%7D+atop+%7Bm_1cdotoverrightarrow%7Bv_1%7Dcdot+m_2cdotoverrightarrow%7Bv_2%7D%3D0%3B%7D+right.+%5C "left { {{overrightarrow{p_1}-overrightarrow{p_2}'=overrightarrow{p_1}'} atop {E_1=E_1'+E_2'}} atop {overrightarrow{p_1}cdotoverrightarrow{p_2}=0} right. left { {{m_1cdotoverrightarrow{v_1}-m_2cdotoverrightarrow{v_2}'=m_1cdotoverrightarrow{v_1}'} atop {m_1cdot v_1^2=m_1cdot v_1'^2+m_2cdot v_2'^2}} atop {m_1cdotoverrightarrow{v_1}cdot m_2cdotoverrightarrow{v_2}=0;} right. \")
![left { {{m_1^2v_1^2-2m_1overrightarrow{v_1}cdot m_2overrightarrow{v_2}'+m_2^2v'_2^2=m_1^2v'_1^2} atop {m_1v_1^2=m_1v1_1^2+m_2v'_2^2}} atop {2m_1overrightarrow{v_1}cdot m_2overrightarrow{v_2}'=0} right.;\](https://tex.z-dn.net/?f=+left+%7B+%7B%7Bm_1%5E2v_1%5E2-2m_1overrightarrow%7Bv_1%7Dcdot+m_2overrightarrow%7Bv_2%7D%27%2Bm_2%5E2v%27_2%5E2%3Dm_1%5E2v%27_1%5E2%7D+atop+%7Bm_1v_1%5E2%3Dm_1v1_1%5E2%2Bm_2v%27_2%5E2%7D%7D+atop+%7B2m_1overrightarrow%7Bv_1%7Dcdot+m_2overrightarrow%7Bv_2%7D%27%3D0%7D+right.%3B%5C+ "left { {{m_1^2v_1^2-2m_1overrightarrow{v_1}cdot m_2overrightarrow{v_2}'+m_2^2v'_2^2=m_1^2v'_1^2} atop {m_1v_1^2=m_1v1_1^2+m_2v'_2^2}} atop {2m_1overrightarrow{v_1}cdot m_2overrightarrow{v_2}'=0} right.;\")
![left { {{m_1^2v_1^2+m_2^2v'_2^2=m_1^2v'_1^2} atop {m_1v_1^2=m_1v'_1^2+m_2v'_2^2}} right.;\
v_1=V; v'_2=frac{V}2;\
m_1=3kg; m_2-?\
left { {{m_1^2V^2+frac14m_2^2V^2=m_1^2v'_1^2} atop {m_1V^2=m_1v'_1^2+frac14m_2V^2|times m1}} right.;\
left { {{m_1^2V^2-m_1^2v'_1^2=-frac14m_2^2V^2} atop {m_1^2V^2-m_1^2v'_1^2=frac14m_1m_2V^2}} right.;\
m_1^2V^2-m_1^2v'_1^2-m_1^2V^2+m_1^2v'_1^2=frac14m_1m_2V^2+frac14m_2^2V^2;\
frac14V^2(m_1m_2+m_2^2)=0;\
m_2=3kg](https://tex.z-dn.net/?f=+left+%7B+%7B%7Bm_1%5E2v_1%5E2%2Bm_2%5E2v%27_2%5E2%3Dm_1%5E2v%27_1%5E2%7D+atop+%7Bm_1v_1%5E2%3Dm_1v%27_1%5E2%2Bm_2v%27_2%5E2%7D%7D+right.%3B%5C+%0Av_1%3DV%3B+++v%27_2%3Dfrac%7BV%7D2%3B%5C%0Am_1%3D3kg%3B+++m_2-%3F%5C%0A+left+%7B+%7B%7Bm_1%5E2V%5E2%2Bfrac14m_2%5E2V%5E2%3Dm_1%5E2v%27_1%5E2%7D+atop+%7Bm_1V%5E2%3Dm_1v%27_1%5E2%2Bfrac14m_2V%5E2%7Ctimes+m1%7D%7D+right.%3B%5C+%0A+left+%7B+%7B%7Bm_1%5E2V%5E2-m_1%5E2v%27_1%5E2%3D-frac14m_2%5E2V%5E2%7D+atop+%7Bm_1%5E2V%5E2-m_1%5E2v%27_1%5E2%3Dfrac14m_1m_2V%5E2%7D%7D+right.%3B%5C+%0A+m_1%5E2V%5E2-m_1%5E2v%27_1%5E2-m_1%5E2V%5E2%2Bm_1%5E2v%27_1%5E2%3Dfrac14m_1m_2V%5E2%2Bfrac14m_2%5E2V%5E2%3B%5C%0Afrac14V%5E2%28m_1m_2%2Bm_2%5E2%29%3D0%3B%5C%0Am_2%3D3kg "left { {{m_1^2v_1^2+m_2^2v'_2^2=m_1^2v'_1^2} atop {m_1v_1^2=m_1v'_1^2+m_2v'_2^2}} right.;\
v_1=V; v'_2=frac{V}2;\
m_1=3kg; m_2-?\
left { {{m_1^2V^2+frac14m_2^2V^2=m_1^2v'_1^2} atop {m_1V^2=m_1v'_1^2+frac14m_2V^2|times m1}} right.;\
left { {{m_1^2V^2-m_1^2v'_1^2=-frac14m_2^2V^2} atop {m_1^2V^2-m_1^2v'_1^2=frac14m_1m_2V^2}} right.;\
m_1^2V^2-m_1^2v'_1^2-m_1^2V^2+m_1^2v'_1^2=frac14m_1m_2V^2+frac14m_2^2V^2;\
frac14V^2(m_1m_2+m_2^2)=0;\
m_2=3kg")
индексы 1 и 2 первый и второй шар
без штриха до взаимодействия(удара), а со штрихом после
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