Two bodies of masses ${m_1}$ and ${m_2}$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is
Two bodies of masses ${m_1}$ and ${m_2}$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is
${\left[ {2G\frac{{({m_1} - {m_2})}}{r}} \right]^{1/2}}$
${\left[ {\frac{{2G}}{r}({m_1} + {m_2}} \right]^{1/2}}$
${\left[ {\frac{r}{{2G({m_1}{m_2})}}} \right]^{1/2}}$
${\left[ {\frac{{2G}}{r}{m_1}{m_2}} \right]^{1/2}}$
Two bodies of masses ${m_1}$ and ${m_2}$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is
Let velocities of these masses at $r$ distance from each other be ${v_1}$ and ${v_2}$ respectively. By conservation of momentum ${m_1}{v_1} - {m_2}{v_2} = 0$
$ \Rightarrow \,{m_1}{v_1} = {m_2}{v_2}$… (i)
By conservation of energy change in $P.E.$ = change in $K.E.$
$\frac{{G{m_1}{m_2}}}{r} = \frac{1}{2}{m_1}v_1^2 + \frac{1}{2}{m_2}v_2^2$
$ \Rightarrow \,\frac{{m_1^2v_1^2}}{{{m_1}}} + \frac{{m_2^2v_2^2}}{{{m_2}}} = \frac{{2\,G{m_1}{m_2}}}{r}$ …(ii)
On solving equation (i) and (ii) ${v_1} = \sqrt {\frac{{2\,Gm_2^2}}{{r({m_1} + {m_2})}}} $ and ${v_2} = \sqrt {\frac{{2\,Gm_1^2}}{{r({m_1} + {m_2})}}} $
$\therefore \,{v_{{\rm{app}}}} = \,|\,{v_1}\,|\, + \,|\,\,{v_2}\,|\, = \,\sqrt {\frac{{2G}}{r}({m_1} + {m_2})} $
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