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Question Number 86886 by Rio Michael last updated on 01/Apr/20

Commented by Rio Michael last updated on 01/Apr/20

$$\mathrm{A}\mathrm{smooth}\mathrm{sphere}\mathrm{S}\mathrm{of}\mathrm{mass}\lambda m,\mathrm{moving}\mathrm{with}\mathrm{speed}\lambda u\mathrm{collides}\mathrm{with}$$$$\mathrm{a}\mathrm{smooth}\mathrm{sphere}\mathrm{T}\mathrm{of}\mathrm{equal}\mathrm{radius}\mathrm{but}\mathrm{mass}m\mathrm{moving}\mathrm{with}\mathrm{speed}u.$$$$\mathrm{At}\mathrm{impact},\mathrm{the}\mathrm{direction}\mathrm{of}\mathrm{motion}\mathrm{of}\mathrm{each}\mathrm{sphere}\mathrm{makes}\mathrm{an}\mathrm{angle}\theta$$$$\mathrm{with}\mathrm{the}\mathrm{line}\mathrm{of}\mathrm{centre}\mathrm{as}\mathrm{shown}\mathrm{above}.$$$$\mathrm{Given}\mathrm{that}\mathrm{the}\mathrm{coefficient}\mathrm{of}\mathrm{restitutions}\mathrm{betweethe}\mathrm{spheres}\mathrm{is}\frac{1}{2},\mathrm{and}\mathrm{after}$$$$\mathrm{impact},\mathrm{the}\mathrm{directionof}\mathrm{S}\mathrm{cames}\mathrm{an}\mathrm{angle}\mathrm{of}2\theta \mathrm{with}\mathrm{the}\mathrm{line}\mathrm{of}\mathrm{centres},\mathrm{show}\mathrm{that}$$$$\left(\mathrm{i}\right)\mathrm{the}\mathrm{speed}\mathrm{of}\mathrm{S}\mathrm{along}\mathrm{the}\mathrm{along}\mathrm{the}\mathrm{line}\mathrm{of}\mathrm{centre}\mathrm{is}\frac{1}{2}u(2\lambda -3)\cos \theta$$$$\left(\mathrm{ii}\right)1+{\tan }^2\theta =\frac{3}{\lambda }$$$$\left(\mathrm{iii}\right)0<\lambda <3$$

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