Two particles are projected simu ltaneously from points A and B respectively and they move in same plane. Find.

(a) the separation when they are closest to each other.
(b) the time elapsed to come closest to each other.
(c) condition that they collide in air.
(d) initial velocity of approch between the particles.
(e) initial angu lar velocity of first particle w.r.t. second particle.

Explanation

Velocity of 1st particle at any instant 't'
`vecv_(1)=u_(1)cos theta_(1)hati+(u_(2)sin theta_(2)-gt)hatj` ........(i)
Velocity of 2nd particle at the same instant 't'
`vec v_(2)=u_(2) cos theta_(2)(-hati)+(u_(2)sin theta_(2)-gt)hatj`...... (ii)
`therefore `velocity of 1st particle w.r.t. 2nd particle
`vecv_(12)=vecv_(1)-vecv_(2)=(u_(1)cos theta_(1)+u_(2)cos theta_(2))hati`
`+(u_(1) sin theta_(1)-u_(2)sin tehta_(2))hatj` .........(iii)
`tan phi=(u_(1)sin theta_(1)-u_(2)sin theta_(@))/(u_(1)cos theta_(1)+u_(2) cos theta_(2))` .........(iv)

(a) Separation when they are closest is BV=`l sin phi`
(b) `t=(AC)/(|vecv_(12)|)=(l cos phi)/(|vecv_(12)|)`
`|vecv_(12)|=sqrt((u_(1)cos theta_(1)+u_(2)cos theta_(2))^(2)+(u_(1)sin theta_(1)-u_(2)sin theta_(2))^(2))`
(c) If they collide then BC (minimum speration)=0
i.e., `phi=0`
`therefore tan phi=0`
`rArr u_(1)sin theta_(1)=u_(@) sin theta_(2)` from equation (iv)
(d) Velocity of approach
`=u_(1) cos theta_(1)+u_(2)cos theta_(2)`

(e) `omega_(12)=omega_(AB)=(v_(12)sin phi)/(l)`
`omega_(12)=(u_(1)sin theta_(1)-u_(2)sin theta_(2))/(l)`