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Question: 15 the brachistochrone quotshortest timequot problem posed by bernoulli...

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( 15 The brachistochrone (shortest time) problem, posed by Bernoulli in 1696. states that a particle of mass m starts at a point Pi(x1 with speed V and moves under gravity (acting in the negative y direction) along a curve y f(x) to a point Ps(x Ye), where yi > ys and x >x and requires the curve along which the elapsed time T is minimum. (a) If v is the speed and s the distance travelled at time t, use the relations dt -dslv and mv + mgy constant to obtain the formulation V2 dx minimuma y1 where y(x1)-Y1 and y(%)-2. (b) Deduce the relation dx dy, -y-ti sin2 ф/2, show that if a minimizing curve exists, V2 1 where c is a constant. (C) By writing then it can be defined by parametric equations of the form where c, and c, are constants to be determined such that P, and Pa are on the curve. [The extremals are cycloids. It is known that c1 and c2 can be uniquely deter- mined so that P, and P2 are on the same arch of one such cycloid and that the corresponding arc P,P2 then truly minimizes T.]
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