A hunter aims his gun and fires a bullet directly at a monkey on a tree. At the instant the bullet leaves the barrels of the gun, the monkey drops. Will the bullet hit the monkey. Substantiate you answer with proper reasoning.

Explanation

Let the monkey stationed at A, be fired with a gun from O with a velocity u at an angle `theta` with the horizontal direction OX. Draw AC perpendicular to OX. Let the bullet cross the vertical line AC at B after time t and coordinates. Of B be (x,u) w.r.t origin O.

`t=(OC)/(u cos theta)=x/(u cos theta)`.............i
In `DeltaOAC, AC=OCtan theta=x tan theta`.........ii
Clearly CB=y=the vertical distance travelled by the bullet in time t
Taking motion of the bullet from O to B along Y axis we have
`y_(0)=0,y=y,u_(y)=u sin theta,a_(y)=-g,t=t`
As `y=y_(0)+u_(y)t+1/2a_(y)t^(2)`
`:.y=theta+u sin thetat+1/2(-t)t^(2)=u sin thetat-1/2"gt"^(2)`.......iii
`:.AB=AC-BC=x tan theta-y`
`=x tan theta-(usin thetat-1/2"gt"^(2))=xtan theta(u sin thetaxx x/(u cos theta)-1/2"gt"^(2))`.........[from (i)]
`x tan theta-x tan theta=1/2"gt"^(2)=1/2"gt"^(2)`
It means the bullet will pass through the point B on vertical line AC at a vertical distance `1/2"gt"^(2)` below point A.
The distance through which the monkey falls vertically in time `t=1/2"gt"^(2)=AB`
It means the bullet and monkey will pass through the point B simultaneously. Therefore, the bullet wil hit the monkey.