Two particles P and Q are projected from two points in the same plane as shown. Find the condition for collision of two paricles.

Explanation

Acceleration of both the particles w.r.t ground is ` vecg` .Now w.r.t ground velocity of particle P.
`vecu_(p) = ( u_(1) cos theta_(1)) hati + ( u_(1) sin theta_(1))hatj`
acceleration of particle P
`veca_(p) = - ghatj`
simiarly
`u_(Q) = - ( u_(2) cos theta_(2))hati + ( u_(2) sin theta_(2))hatj and veca_(Q) = - ghat`
Now relative velocity and acceleration of P w.r.t Q
` veca_(PiQ)= vecu_(p) - vecu_(Q)`
`= ( u_(1) cos theta_(1) + u_(2) cos theta_(2)hati + ( u_(1) sin theta_(1) - vecu_(2) sin theta_(2))hatj`
`veca_(PIQ) = vec0`
As one projectile moves on a straight line w.r.t another projectile, here ` vecu_(PIQ)` is the velocity of Q as seen by P.
Now, if they colider after time t then
` X_(rel) = U_(x_(re)) t`
` Rightarrow d = u_(1) (costheta_(1) + u_(2) cos theta_(2))t`
` and y_(rel) = U_(y rel).t`
` h_(2) -h_(1) = ( u_(1) sintheta_(1) - u_(2) sintheta_(2)) t`
from equation (i) and (ii)
` (h^(2) -h^(1))/d = ( u_(1) sin tehta_(1) - u_(2) sin theta_(2))/(u_(1) cos teheta_(1) + u_(2) cos theta_(2))`
This is the condition for two particles to collide.