A) \[\sqrt{\frac{2Gd}{M+m}}\]
B) \[\sqrt{\frac{2G(M+m)}{d}}\]
C) \[\sqrt{\frac{2GMm}{(M+m)d}}\]
D) \[\sqrt{\frac{2Gm}{d}}\]
Correct Answer: B
Solution :
1st method- Let us say the spheres are moving with velocities \[{{v}_{1}}\And {{v}_{2}}\] when they are at a separation of d. Then from momentum conservation, \[m{{v}_{1}}=M{{v}_{2}}\] From energy conservation, \[\left( \frac{mv_{1}^{2}}{2}+\frac{Mv_{1}^{2}}{2} \right)-0=\left[ -\frac{GMm}{d}-0 \right]\] After solving above equation we get \[{{v}_{1}}+{{v}_{2}}=\sqrt{\frac{2G(M+m)}{d}}\] II2nd Method : Using C frame or reduced mass concept. Let \[{{v}_{r}}\] be the relative velocity when they are at a separation of d. then,\[\left( \frac{mM}{m+M} \right)\frac{v_{r}^{2}}{2}=-\left[ -\frac{GMm}{d}-0 \right]\] \[\Rightarrow \]\[{{v}_{r}}=\sqrt{\frac{2G(M+m)}{d}}\]You need to login to perform this action.
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