Paper 2, Section I, C

Dynamics and Relativity | Part IA, 2020

A particle PP with unit mass moves in a central potential Φ(r)=k/r\Phi(r)=-k / r where k>0k>0. Initially PP is a distance RR away from the origin moving with speed uu on a trajectory which, in the absence of any force, would be a straight line whose shortest distance from the origin is bb. The shortest distance between PP 's actual trajectory and the origin is pp, with 0<p<b0<p<b, at which point it is moving with speed ww.

(i) Assuming u22k/Ru^{2} \gg 2 k / R, find w2/kw^{2} / k in terms of bb and pp.

(ii) Assuming u2<2k/Ru^{2}<2 k / R, find an expression for PP^{\prime} 's farthest distance from the origin qq in the form

Aq2+Bq+C=0A q^{2}+B q+C=0

where A,BA, B, and CC depend only on R,b,kR, b, k, and the angular momentum LL.

[You do not need to prove that energy and angular momentum are conserved.]

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