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ভরবেগের সংরক্ষণশীলতা ও শক্তির সংরক্ষণশীলতা সূত্র থেকে v1 ও v2 নির্ণয়
Price: Free Course Post By: Mehedi সর্ব-শেষ হাল-নাগাদ: 05 Sunday 2020

ভরবেগের সংরক্ষণশীলতা ও শক্তির সংরক্ষণশীলতা সূত্র দুটি যথাক্রমে এভাবে লিখতে পারি,

m_{1}(u_{1}-v_{1})=m_{2}(v_{2}-u_{2}) ...................(1)

m_{1}(u_{1}^{2}-v_{1}^{2})=m_{2}(v_{2}^{2}-u_{2}^{2}) ...................(2)



(2) নং কে (1) নং দ্বারা ভাগ করে পাই,

frac{m_{1}(u_{1}^{2}-v_{1}^{2})}{m_{1}(u_{1}-v_{1})}=frac{m_{2}(v_{2}^{2}-u_{2}^{2})}{m_{2}(v_{2}-u_{2})}

বা, frac{m_{1}(u_{1}+v_{1})(u_{1}-v_{1})}{m_{1}(u_{1}-v_{1})}=frac{m_{2}(v_{2}+u_{2})(v_{2}-u_{2})}{m_{2}(v_{2}-u_{2})}

বা, u_{1}+v_{1}=v_{2}+u_{2}

বা, v_{2}=u_{1}+v_{1}-u_{2} ....................(3)



ভরবেগের সংরক্ষণশীলতা সূত্র হতে আমরা পাই,

m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}

বা, m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}(u_{1}+v_{1}-u_{2}) [ (3) নং এর মান বসিয়ে ]

বা, m_{1}u_{1}+m_{2}u_{2}={color{Red} m_{1}v_{1}}+m_{2}u_{1}+{color{Red}m_{2}v_{1}}-m_{2}u_{2}

বা, {color{Red} m_{1}v_{1}+m_{2}v_{1}} = m_{1}u_{1}+m_{2}u_{2}-m_{2}u_{1} + m_{2}u_{2}

বা, v_{1}(m_{1}+m_{2})= (m_{1}-m_{2})u_{1} + 2m_{2}u_{2}

বা, {color{Blue} v_{1}= frac{(m_{1}-m_{2})u_{1} + 2m_{2}u_{2}}{(m_{1}+m_{2})}}



(3) নং এ আমরা v1 এর এই মান বসিয়ে পাই,

v_{2}=u_{1}-u_{2}+v_{1}

বা, v_{2}=u_{1}-u_{2}+ frac{(m_{1}-m_{2})u_{1} + 2m_{2}u_{2}}{(m_{1}+m_{2})}

বা, v_{2}=u_{1}-u_{2}+ frac{(m_{1}-m_{2})u_{1} }{(m_{1}+m_{2})}+ frac{ 2m_{2}u_{2}}{(m_{1}+m_{2})}

বা, v_{2}=u_{1}+ frac{(m_{1}-m_{2})u_{1} }{(m_{1}+m_{2})}+ frac{ 2m_{2}u_{2}}{(m_{1}+m_{2})}-u_{2}

বা, v_{2}=u_{1}[1+ frac{(m_{1}-m_{2}) }{(m_{1}+m_{2})}]+ u_{2}[frac{ 2m_{2}}{(m_{1}+m_{2})}-1]

বা, v_{2}=u_{1}[ frac{m_{1}+m_{2}+m_{1}-m_{2} }{m_{1}+m_{2}}]+ u_{2}[frac{ 2m_{2}-m_{1}-m_{2}}{m_{1}+m_{2}}]

বা, v_{2}=u_{1}[ frac{2m_{1} }{m_{1}+m_{2}}]+ u_{2}[frac{ m_{2}-m_{1}}{m_{1}+m_{2}}]

বা, v_{2}= frac{ (m_{2}-m_{1})u_{2}}{m_{1}+m_{2}}+ frac{2m_{1} u_{1}}{m_{1}+m_{2}}

বা, {color{Blue} v_{2}= frac{ (m_{2}-m_{1})u_{2}+2m_{1} u_{1}}{m_{1}+m_{2}}}