The radii of a planet and its satellite are 2r and r and their densities are $ρ$ and $2 ρ$ respectively. Their centres are separated by a distance d. The minimum speed with which a body should be projected from the mid point of the line joining their centres so that the body escapes to infinity is (G-universal gravitational constant)

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$\sqrt{\frac{10 {Gπr}^{3} ρ}{3 d}}$

$\frac{1}{4} [\sqrt{\frac{10 G π r^{3} ρ}{3 d}}]$

$2 [\sqrt{\frac{10 G π r^{3} ρ}{d}}]$

$4 [\sqrt{\frac{10 G π r^{3} ρ}{3 d}}]$

answer is A.

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Detailed Solution

As a body of mass 'm' is projected from mid point of line joing the two masses m₁, m₂
$\begin{array}{l} (KE + PE)_{i} = (KE + PE)_{f} \\ \frac{1}{2} {mv}_{e}^{2} - \frac{{GM}_{1} m}{d / 2} - \frac{{GM}_{2} m}{d / 2} = 0 \\ v_{e}^{2} = \frac{4 G (M_{1} + M_{2})}{d} = \frac{4 G (\frac{4 π}{3} (2 r)^{3} ρ + \frac{4 π}{3} (r)^{3} 2 ρ)}{d} \\ v_{e} = 4 \sqrt{\frac{10 {Gπr}^{3} ρ}{3 d}} \end{array}$

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