Consider a non conduting ring of radius r and mass m and a particle of same mass, both at rest in free space. The particle is on the axis of the ring and far away from the ring. An amount Q of positive charge is uniformly distributed on the ring and the particle is given a positive charge q. The particle is imparted a velocity v towards the centre of the ring. Consider the consequences given in the columns and answer the following question

`{:(,"Column-I",,"Column-II",,"Column-III"),((I),"Maximum speed of the ring is v",(i),"Minimum speed of the particle is"v//2,(P),"Final speed of the ring and particle is v and zero respectively".),((II),"Maximum speed of the ring is" v//2,(ii),"Minimum speed of the particle is zero",(Q),"Final speed of the ring and particle is zero and v respectively".),((III),"Maximum speed of the ring is"(v)/(2)[1+sqrt(1-(Qq)/(pi epsilon_(0)mrv^(2)))],(iii),"Minimum speed of the particle is"(v)/(2)[1+sqrt(1-(Qq)/(pi epsilon_(0)mrv^(2)))],(R ),"Final speed of the ring and particle is v/4 and3v/4 respectively".),((IV),"Maximum speed of the ring is" (v)/(2)[1-sqrt(1-(Qq)/(pi epsilon_(0)mrv^(2)))],(iv),"Minimum speed of the particle is" (v)/(2)[1-sqrt(1-(Qq)/(pi epsilon_(0)mrv^(2)))],(S),"Final speed of the ring and particle is v/2 for both".):}`
Which of following options is the correct is the representation if`v=sqrt((Qq)/(2piepsilon_(0)mr)`

Options

A. `(II)(i)S`
B. `(I)(ii)P`
C. `(III)(iv)R`
D. `(IV)(iii)Q`

Answer

B

Explanation

Let v be the velocity of particle for which both moves together finally

`p_(i)=p_(f)`
`mv=2mv' implies v'=(v)/(2)" "….(i)`
`E_(i)=E_(f)`
`(1)/(2)mv^(2)((1)/(2)mv^('2))xx2-(kQq)/(r).....(ii)`
`v=sqrt((Qq)/(piepsilon_(0)mr))`
If `vltsqrt((Qq)/(piepsilon_(0)mr))` then velocity of particle continously decrease and that of ring continously increase. Final velocity of particle becomes zero and that of ring becomes v
If `vgtsqrt(Qq)/(piepsilon_(0)mr)` then particle crosses the ring and again it's velocity start increasing and finally becomes v and that of ring becomes zero.
`p_(i)=p_(f)impliesmv=mv_(1)+mv_(2)....(i)`
`(1)/(2)mv^(2)=(1)/(2)mv_(1)^(2)+(1)/(2)mv_(2)^(2)+(kQq)/(r)....(ii)`
Minimum speed of particle
`v_(1)=(v)/(2)[1+sqrt(1-(Qq)/(piepsilon_(0)mrv^(2))]]`

Maximum speed of ring
`v_(2)=(v)/(2)[1-sqrt(1-(Qq)/(piepsilon_(0)mrv^(2))]]`