Two cacar starts together from the asme point and move along two straight lines inclined at an angle ` theta`, one moving with velocity (u) and the other from rest with a uniform acceleration (a). Show that the least relative velocity between then is ` u sin theta` and it occurs after a time ` t= ( u cos theta) //a`.

Explanation

Let the first car be moving with uniform velocity (u) along (OA) and the second car be moving with uniform acceleration (a) along ` OB`, Fig. 2. (c ) . 98.
After (t) seconds the velocity of second car,
`v= at ` ltbRgt Relative velocity of first car w.r..t the second
car is ` vec v_r = vec u- vec v = vec u + ( - vec v)`
` vec (OA) + vec (OC) = vec (OD)`
.
Magniude of ` vec v_r` is given by
` v_r^2 =u^2 + v^2+ 2 uv cos ( 180^2 - theta)`
`v_r = [ u^2 _a^2 t^2 -2 u at cos theta ] ^(1//2)`
` v_r ` is minimum if ` ( dv_r) /(dt) =0`
(dv_r)/(dt) = 1/2 ( u^2+a^2t^2-2 u at cos theta) ^(-1//2)`
`xx 9 2 A^2 t- 2 u a cos theta)=0`
Hence, ` 2 a^2 t- 2 u a u a cos theta =0`
or ` t= ( u cos theta)/a`
` :. v_r = [ u^2 +a^2 xx ( u^2 cos^2 theta)/a^2 -2ua ( ucos theta)/a xx cos theta]^(1//2)`.
` [ u^2 -u^2 cos^2 hteta]^(1//2)`
`= u sin theta (Proved)`.