A small body of mass is projected with a velocity just sufficient to it reach from the surface of a planet -of radius 2R and mass 3M- to the surface of another planet -of radius R and mass M- The distance between the centers of the two spherical planets is 6R- The distance of the body from the center of bigger planet is -x- any moment- During the journey- the distance x where the speed of the body is maximum- Assume motion of body along the line joining centres of planets-

Question

Toppr Admin · Accepted Answer

The force on the body at a distance x would be

$F = \frac{G 3 M m}{x^{2}} - \frac{G M m}{{(6 R - x)}^{2}} = G M m (\frac{3}{x^{2}} - \frac{1}{{(6 R - x)}^{2}})$

It is obvious that the velocity of the body will first devrease and then increase. Hence the maximum velocity will either be at $x = 2 R$ or $x = 5 R$ .

Conserving energy,

$\frac{1}{2} m v_{1}^{2} - \frac{3 G M m}{2 R} - \frac{G M m}{4 R} = \frac{1}{2} m v_{2}^{2} - \frac{G M m}{R} - \frac{3 G M m}{5 R}$

$\frac{1}{2} m {(v}_{1}^{2} - v_{2}^{2}) = \frac{G M m}{R} (\frac{3}{2} - \frac{3}{5} + \frac{1}{4} - 1) > 0$

Thus, $v_{1} > v_{2}$

Hence, answer is $2 R$