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# Question on central force

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A particle of unit mass is projected with speed (see attached) at right angles to the radius vector at a distance a from a fixed point O and is subjected to a force F(r) = μ/r^2, with μ a positive constant. Show that the particle moves on a path which passes the centre at a distance of 3a at its maximum.

See attached

https://brainmass.com/math/algebraic-geometry/question-central-force-538260

#### Solution Preview

You are given mass, m = 1
Initial velocity vi = sqrt(3mu/2a)
Initial distance, ri = a

Let us assume, maximum distance rf = r (to be estimated, and to be proved to equal to 3a)
Let us assume, velocity at maximum distance vf = v [Note at the maximum distance, velocity will be perpendicular to r again)

You can visualize the problem as initially particle at one of the ends of minor axis of elliptical orbit. At maximum distance, the particle is at ...

#### Solution Summary

The solution utilises concept of potential energy and law of conservation of angular momentum to trace the path of particle under central force.

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