d
અહી દરેક પદાર્થ નો પ્રવેગ \({\text{a}}\,\, = \,\,\frac{{{{\text{m}}_{\text{1}}}\, - \,\,{m_2}}}{{{m_1}\, + \,\,{m_2}}}\,g\)

દ્રવ્યમાન કેન્દ્ર નો પ્રવેગ ,\(\,\mathop {{{\text{a}}_{{\text{cm}}}}}\limits^ \to  \, = \,\,\frac{{{m_1}\,\mathop {{a_1}}\limits^ \to  \, + \,\,{m_2}\,\mathop {{a_2}}\limits^ \to  }}{{{{\text{m}}_{\text{1}}}\, + \,\,{m_2}}}\)

બંને પદાર્થ વિરુદ્ધ દિશા માં ગતિ કરતાં હોવા થી, \(\mathop {{{\text{a}}_{\text{1}}}}\limits^ \to  \,\, = \,\,{\text{ - }}\,\,\mathop {{{\text{a}}_{\text{2}}}}\limits^ \to  \,\, = \,\,{\text{a}}\,\) 

\(\therefore \,\,{{\text{a}}_{{\text{cm}}}}\, = \,\,\frac{{{m_1}a\,\, - \,\,{m_2}a}}{{{m_1}\, + \,\,{m_2}}}\,\,\,\, = \,\,a\,\left( {\frac{{{m_1}\, - \,\,{m_2}}}{{{m_1}\,\, + \,\,{m_2}}}} \right)\,\)

\( = \,\,\left( {\frac{{{m_1}\, - \,\,{m_2}}}{{{m_1}\, + \,\,{m_2}}}} \right)\,\,\, \cdot \,g\,\,\,\left( {\frac{{{m_1}\, - \,\,{m_2}}}{{{m_1}\, + \,\,{m_2}}}} \right) = \,\,{\left( {\frac{{{m_1}\, - \,\,{m_2}}}{{{m_1}\, + \,\,{m_2}}}} \right)^2}\,g\)