Match the List-I with List-II (O is the point object shown in diagram)

Options

A. `{:(P,Q,R,S),(3,1,4,2):}`
B. `{:(P,Q,R,S),(1,2,3,2):}`
C. `{:(P,Q,R,S),(3,1,3,2):}`
D. `{:(P,Q,R,S),(1,4,2,4):}`

Answer

C

Explanation

(Q) Let angle of incidence is `theta`
Velocity of object `v_(0)=0`
velocity of mirror `v_(m) =-v_(i)`
velocity of object w.r.t. mirror
`v_(om)=v_(o)-v_(m)=+vi`
`v_(IM) = v_(I)-v_(m)`
or `v_(I) =v_(IM)+v_(m)`
`V_(IM) = -vcos2theta hat(i) - vsin 2thet hat(j)`
`v_(I)=(-vcos2theta-v)hat(i)-v sin 2theta hat(j)`
`theta =30^(0),v =sqrt(3)m//s`
`|v_(I)|=3 m//s`
(R) Let angle of incidence is `theta`
`r_(I)=(-xcos2theta)hat(i)+(-x sin 2 theta)hat(j)`
`(d)/(dt)(r_(I))=V_(I),(d theta)/(dt)=omega`
`v_(1)=[-x xx(-sin2 theta)xx2.(d theta)/(dt)hat(i)]-x.cos2 theta xx2.(d theta)/(dt)hat(j)`
`V_(I)=2omegax sin 2 theta hat(i) - 2 omega x cos 2 theta hat(j) `
Put the values. `|V_(I)|=2m//s`
(S) Let angle of incidence is `theta` Position vector of image
`=r_(I)=(-x cos 2theta hat(i))+(-x sin 2 theta)hat(j)`
`(d)/(dt)(r_(I))=v_(I),(dx)/(dt)=v, (d theta)/(dt)=omega`
`(d)/(dt)(r_(I))=[(-dx)/(dt). cos2 theta-x(-sin2 theta).2(d theta)/(dt)]hat(i)`
`+[((-dx)/(dt))sin 2 theta-x.cos2 theta xx2.(d theta)/(dt)]hat(j)`
`V_(I)=[-Vcos2 theta+2omegax sin 2 theta]hat(i)+[-V sin 2 theta-2omega x cos 2 theta]hat(j)`
Put the values of `theta, omega` & v
`|V_(I)|=a=(31)/(9)cm//s`