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