A smooth ring is kept on a smooth horizontal surface. From a point P of the ring a particle is projected at an angle α to the radius vector at P. If e is the coefficient of restitution between the ring and the particle and the particle will return to the point of projection after two reflections then $$1e+1e2+1e3$$ is equal $$to__$$

Question

A  smooth ring is kept on a smooth horizontal surface. From a point P of the  ring a particle is projected at an angle α      to the radius  vector at P. If e is the coefficient of restitution between the ring and the  particle and the particle will return to the point of projection after two  reflections then $$1e+1e2+1e3$$      is equal $$to__$$

Infinity Learn · Accepted Answer

Let u be the velocity of projection at P. We can find the velocity of rebound at point Q fromtanβ$$=usin$$αeucosα$$=$$$$tan$$αeSimilarly, $$tan$$γ$$=$$$$tan$$β$$e=$$$$tan$$α$$e2$$ $$2$$α$$+2$$β$$+2$$γ$$=$$π α$$+$$β$$+$$γ$$=$$π$$2$$ α$$+$$β$$=$$π$$2-$$γ $$tan$$α$$+$$$$tan$$β$$1-$$$$tan$$α$$tan$$β$$=$$$$cot$$γ$$=e2tan$$α $$tan$$α$$+$$$$tan$$α$$e1-$$$$tan$$α$$tan$$α$$e=e2tan$$α on  solving we get  $$cot2$$α$$=e+e2+1e3=1e+1e2+1e3$$

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