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Two massive particles of mass \mathrm{m_1 \: and \: m_2} are released from rest from a very large distance. Find the speeds of the particles, when their distance of separation is \mathrm{r} .
 

Option: 1

\mathrm{m_1 \sqrt{\frac{G}{\left(m_1+m_2\right) r}}}

 


Option: 2

\mathrm{m_1 \sqrt{\frac{2 G}{\left(m_1+m_2\right) r}}}
 


Option: 3

\mathrm{m_1 \sqrt{\frac{G}{2\left(m_1+m_2\right) r}}}
 


Option: 4

\mathrm{2 m_1 \sqrt{\frac{G}{\left(m_1+m_2\right) r}}}


Answers (1)

best_answer

Since the net force acting on the system is equal to, \mathrm{F=\vec{F}_1+\vec{F}_2=0}, the momentum of the system at this instant, is equal to its initial momentum, that is zero because the system is released from rest.

\mathrm{\Rightarrow\left|\mathrm{m}_1 \overrightarrow{\mathrm{v}}_1+\mathrm{m}_2 \overrightarrow{\mathrm{v}}_2\right|=0}

\mathrm{\Rightarrow \mathrm{m}_1 \mathrm{v}_1=\mathrm{m}_2 \mathrm{v}_2 (numerically)}           ..........(1)

Conservation of energy yields

\mathrm{ \mathrm{E}_1=\mathrm{E}_2 }

\mathrm{ \mathrm{~K}_1+\mathrm{U}_1=\mathrm{K}_2+\mathrm{U}_2 }

Since the system is released from rest, \mathrm{ K_1=0 }. Since the particles are released from very large distance, \mathrm{ U_1=0}

\mathrm{ \Rightarrow \mathrm{K}_2+\mathrm{U}_2=0 }

\mathrm{ \Rightarrow\left(\frac{1}{2} \mathrm{~m}_1 \mathrm{v}_1^2+\frac{1}{2} \mathrm{~m}_2 \mathrm{v}_2^2\right)+\left(-\frac{G \mathrm{~m}_1 \mathrm{~m}_2}{\mathrm{r}}\right)=0 }   ..........(2)

Eliminating \mathrm{ v_2 } from (2) by putting \mathrm{v_2=\frac{m_1 v_1}{m_2}} from (1), we obtain,

\mathrm{ \frac{1}{2} m_1 v_1^2+\frac{1}{2} m_2\left(\frac{m_1 v_1}{m_2}\right)^2=\frac{G m_1 m_2}{r} }

\mathrm{ \Rightarrow \frac{m_1 v_1^2}{2}\left[1+\frac{m_1}{m_2}\right]=\frac{G m_1 m_2}{r} }

\mathrm{ \Rightarrow v_1=\sqrt{\frac{2 G m_2^2}{\left(m_1+m_2\right) r}}=m_2 \sqrt{\frac{2 G}{\left(m_1+m_2\right) r}} }

\mathrm{ \text { Similarly, } v_2=\frac{m_1 v_1}{m_2}=m_1 \sqrt{\frac{2 G}{\left(m_1+m_2\right) r}} . }

Hence option 2 is correct.



 

Posted by

rishi.raj

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