The distance between two moving particles `P` and `Q` at any time is a.If `v_(r)` be their relative velocity and if `u` and `v` be the components of `v_(r)`, along and perpendicular to `PQ`.The closest distance between `P` and `Q` and time that elapses before they arrive at their nearest distance is

Options

A. `(a(v+v_(r)))/v,a(1+v_(r)/v)^(2)`
B. `(av)/((v+v_(r))),a(1+u/v_(2))^(2)`
C. `(av_(r))/v,(av_(r))/u^(2)`
D. `(av)/v_(r),(au)/v_(r)^(2)`

Answer

D

Explanation

Assuming `P` to be at rest, particle `Q` is moving with velocity `v_(r)`,in the direction shown in figure.components of `v_(r)`along and perpendicular to `PQ` are `u` and `v` respectively, In the figure `sin alpha=u/v_(r),cosalpha=u/v_(r)`
The closest distance between the particles is `PR`
`S_(min)=PR=PQ=cos alpha=(a)(v/v_(r))rArrS_(min)=(av)/v_(r)`
Time after which they arrive at their nearest distance is
`t=(QR)/v_(r)=((PQ)sinalpha)/v_(r)=((a)(v/v_(r)))/v_(r)=(au)/v_(r)^(2)`