Two planets of radii (**R**) each, but mass (**M**) and (**4M**) have a center to center separation of (**6R**), as shown. A rocket of mass (**m**) is projected from the surface of planet of mass (**M**) directly towards the center of second planet. Obtain an expression for the minimum speed (**v**) of the rocket so that it reaches the surface of the second planet.

### Step by step derivation

The rocket is acted upon by two mutually opposing gravitational forces of the two planets. The neutral point (N) is defined as the position on O-C line where the two forces cancel each other exactly.

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Once the rocket is projected to neutral point (N), its velocity would be zero and then the gravitational pull of second planet would pull it to its surface. If ON = r, we have,

The neutral point r = -6R does not concern us. Thus ON = r = 2R

It is sufficient to project the rocket with a speed that would enable it to reach N. The mechanical energy at surface of planet M is

At neutral point N, the speed approaches zero. The mechanical energy at N is purely potential energy.

Now applying the principle of conservation of energy,

or

or

**At neutral point speed is zero, but when it strikes the second planet, it is not zero.**

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