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Minimum speed to send a particle from one planet to another

This type of question, asked 2 times: 2021 Paper 16, 2021 Paper 22. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 16

    17
    Two planets AA and BB of masses MM and 9M9M with radii RR and 2R2R respectively are present 8 RR distance away from each other. Minimum velocity with which a particle is projected from surface of A such that it reaches planet B is given by aGM7R\sqrt{\frac{aGM}{7R}}. Value of ' aa ' is

  2. Question 2

    2021 Paper 22

    9
    The masses and radii of the earth and moon are (M1,R1)\left( M_{1},R_{1} \right) and (M2,R2)\left( M_{2},R_{2} \right) respectively. Their centres are at a distance ' rr ' apart. Find the minimum escape velocity for a particle of mass ' mm ' to be projected from the middle of these two masses:

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