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Chhattisgarh PMT Chhattisgarh PMT Solved Paper-2005

  • question_answer
    The masses and radii of the earth and moon are \[{{M}_{1}},\,{{R}_{1}}\] and \[{{M}_{2}},\,{{R}_{2}}\] respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is

    A) \[\sqrt[2]{\frac{G}{a}({{M}_{1}}+{{M}_{2}})}\]  

    B) \[\sqrt[2]{\frac{2G}{a}({{M}_{1}}+{{M}_{2}})}\]

    C) \[\sqrt[2]{\frac{Gm}{a}({{M}_{1}}+{{M}_{2}})}\]              

    D) \[\sqrt[2]{\frac{Gm({{M}_{1}}+{{M}_{2}})}{d({{R}_{1}}+{{R}_{2}})}}\]

    Correct Answer: A

    Solution :

    Condition for escaping of a particle is \[KE=PE\] or            \[\frac{1}{2}m{{v}^{2}}=\frac{G{{M}_{1}}m}{d/2}+\frac{G{{M}_{2}}m}{d/2}\]                 \[{{v}^{2}}=\frac{4G}{d}({{M}_{1}}+{{M}_{2}})\]                 \[v=2\sqrt{\frac{G({{M}_{1}}+{{M}_{2}})}{d}}\]


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