A-particle-of-mass-m-moving-with-a-speed-v-hits-elastically-another-stationary-particle-of-mass-2m-on-a-smooth-horizontal-circular-tube-of-radius-r-The-time-in-which-the-next-collision-will-take-plac

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Question Number 24945 by Tinkutara last updated on 29/Nov/17

$$\mathrm{A}\mathrm{particle}\mathrm{of}\mathrm{mass}m\mathrm{moving}\mathrm{with}\mathrm{a}$$$$\mathrm{speed}v\mathrm{hits}\mathrm{elastically}\mathrm{another}$$$$\mathrm{stationary}\mathrm{particle}\mathrm{of}\mathrm{mass}2m\mathrm{on}\mathrm{a}$$$$\mathrm{smooth}\mathrm{horizontal}\mathrm{circular}\mathrm{tube}\mathrm{of}$$$$\mathrm{radius}r.\mathrm{The}\mathrm{time}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{next}$$$$\mathrm{collision}\mathrm{will}\mathrm{take}\mathrm{place}\mathrm{is}\mathrm{equal}\mathrm{to}$$

Commented by Tinkutara last updated on 30/Nov/17

$$\mathrm{Thank}\mathrm{you}\mathrm{Sir}!$$

Commented by mrW1 last updated on 29/Nov/17

$$sincethehitiswithoutenergylost,$$$$thedepartingvelocityafterthehit$$$$isequaltotheapprochingvelocity$$$$beforethehit,whichisv.$$$$i.e.afterthehitbothparticles$$$$departfromeachotherwithvin$$$$thetube.thetimeafterwhichtheyhit$$$$againishence\frac{2\pi r}{v}.$$

Commented by mrW1 last updated on 29/Nov/17

$$Ithinkit{doesn}^{\prime }tmatterwhichmass$$$$theyhave.$$

Commented by mrW1 last updated on 30/Nov/17

$$thisistheproof:$$$$let{m}_2=\lambda m_1$$$$beforethehit:$$$$particle1:v$$$$particle2:0$$$$afterhit:$$$$particle1:u_1$$$$particle2:u_2$$$$m_{1}v=m_1{u}_1+m_2{u}_2$$$$\Rightarrow v=u_1+\lambda u_2\dots \left(i\right)$$$$\frac{1}{2}{m}_1{v}^2=\frac{1}{2}{m}_1{u}_1^2+\frac{1}{2}{m}_2{u}_2^2$$$$\Rightarrow v^2=u_1^2+\lambda u_2^2\dots \left(ii\right)$$$$\left(i\right)into\left(ii\right)$$$$\Rightarrow v^2=v^2-2\lambda {vu}_2+{\lambda }^2{u}_2^2+\lambda u_2^2$$$$\Rightarrow \lambda u_2[(\lambda +1)u_2-2v]=0$$$$\Rightarrow (\lambda +1)u_2=2v$$$$\Rightarrow u_2=\frac{2}{\lambda +1}\times v$$$$\Rightarrow u_1=v-\lambda \times \frac{2v}{\lambda +1}=\frac{1-\lambda }{\lambda +1}\times v$$$$\Rightarrow \mathrm{\Delta }u=u_2-u_1=\frac{2-1+\lambda }{\lambda +1}\times v=v$$$$i.e.afterhittheparticlesdepartfrom$$$$eachotherwiththesamevelocity$$$$astheyapprochtoeachotherbefore$$$$thehit.$$$$thetimetillnextcollisionishence$$$$\frac{2\pi r}{v}$$

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