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))/(u))^(2)`
B. `(av)/((v+v_(r))),a(1+(u)/(v_(r)))^(2)`
C. `(av_(r))/(v),(av_(r))/(u)`
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^(3))`, in the direction shown in figure, cpmponents of `v_(r)` along and perpendicular to PQ are u and v respectively. In the figure `sin alpha=(u)/(v_(r)) cos alpha=(v)/(v_(r))`

The closed distance between the particles is PR.
`S_("min")=PR= PQ cos alpha=(a) ((V)/(V_(r)) rArr S_("min")=(av)/(v_(r))`
Time after which they arrive at their nearest , distance is