Q138 / 150 · MHT CET 2022 (08 Aug Shift 1) · Physics

The mass and radius of the earth and moon are $M, R$ and $m, r$ respectively. The distance between their centers is $d$. The minimum velocity with which a particle of mass $m_0$ should be projected from the midpoint between them so that it could reach infinity is

  • Correct answer
    A
    $2 \sqrt{\frac{G}{d}(M+m)}$
  • B
    $2 \sqrt{\frac{G m}{d}(M+m)}$
  • C
    $2 \sqrt{\frac{2 G}{d}(M+m)}$
  • D
    $2 \sqrt{\frac{G m(M+m)}{d(R+r)}}$

Solution

The potential energy at the midpoint is $\left(-\frac{G m m_0}{d / 2}-\frac{G M m_0}{d / 2}\right)$ And suppose $\left(\frac{m_0 V^2}{2}\right)$ kinetic energy is required to just send the particle to infinity. On considering energy conservation, i.e., initial total energy is equal to the final total energy: $\begin{aligned} & \left(-\frac{G m m_0}{d / 2}-\frac{G M m_0}{d / 2}\right)+\left(\frac{m_0 V^2}{2}\right)=0 \\ & \Rightarrow V^2=\frac{4 G}{d}(M+m) \\ & \Rightarrow V=2 \sqrt{\frac{G(M+m)}{d}} \end{aligned}$

Hence, the correct answer is option: 'A'.

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