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Minimum speed to escape to infinity from a point between the earth and the moon

The same question, asked 3 times: May 2022 S1*, 3 May 2023 S1*, 16 Apr 2026 S1. Every wording below, oldest first.

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  1. Question 1

    May 2022 S1

    18
    The masses and radii of the moon and the earth are M1,R1M_1, R_1 and M2,R2M_2, R_2 respectively. Their centres are at a distance dd apart. What should be the minimum speed with which a body of mass mm should be projected from a point midway between their centres, so as to escape to infinity?

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  2. Question 2

    3 May 2023 S1

    17
    Earth has mass 'M1M_1' radius 'R1R_1' and for moon mass 'M2M_2' and radius 'R2R_2'. Distance between their centres is 'r'. A body of mass 'M' is placed on the line joining them at a distance r3\frac{r}{3} from the centre of the earth. To project a mass 'M' to escape to infinity, the minimum speed required is

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  3. Question 3

    16 Apr 2026 S1

    109
    The masses and radii of the earth and moon are M1,R1M_1, R_1 and M2,R2M_2, R_2 and respectively, Their centres are at a distance 'd' apart. The minimum speed with which body of mass 'm' should be projected from a distance 2d/32d/3 from the centre of M1M_1 so as to escape to infinity is

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