Two guns situated at the top of a hill of height `10 m` fire one shot each with the same speed `5sqrt(3) m//s` at some interval of time. One gun fires horizontal and the other fores upwards at an angle of `60^(@)` with the horizontal. Two shots collide in air at a poit `P`. Find (i) time-interval between the firing and (ii) coordinates of the point `P`. Take the origin of coordinates system at the foot of the hill right below the muzzle and trajectorise in the `x-y` plane.

Answer

(i) 45 (ii) 2

Explanation

(i) 45 (ii) (2)
(i) Let A stands for trolley and B for ball Relative velocity of B with respect to `A(v_(BA))` should be along OA for the ball to hit the trolley.
Hence will make an angle of `45^@` with positive x-axis
(ii) Let v =speed of ball with respect to surface
`phi=(4 theta)/(3)=4(45^(@))/(3)=60^(@), vecv_(BG)=v cos 60^(@) hati+v sin 60^(@) hatj=v/2 hati+sqrt3/2 v hatj`
`vec_(AG)=(sqrt3-1)hatj, vecv_(BA)=vecv_(BG)-vecv(AG) =v/2hati+(sqrt3/2v-sqrt3+1)hatj`
Since `vecvBa" makes "45^(@)" with x axis "therefore v/2=sqrt3/2 v-sqrt3+1`
`rArr (sqrt-1)v/2=(sqrt3-1) rArr v=2m//s`