A body is projected vertically upward from point A, the top of a tower. It reaches the ground in `t_(1)` sec. If it is projected vertically downwards from A with the same velocity it reaches the ground in `t_(2)` sec. If it falls freely from A , it would reach the ground in ______ .

Options

A. `(t_(1)+t_(2))/(2)`sec
B. `(t_(2)-t_(2))/(2)`sec
C. `t_(1)t_(2)` sec
D. `sqrt(t_(1)t_(2))`sec

Answer

D

Explanation

For three journies of the body
`h=-ut_(1)+(1)/(2)"gt"_(1)^(2) " "implies(h)/(t_(1))=-u+(1)/(2)"gt"_(1)" "cdots(1)`
`h=ut_(2)+(1)/(2)"gt"_(2)^(2), " "implies(h)/(t_(2))=u+(1)/(2)"gt"_(2)" "cdots(2)`
`h= 0+(1)/(2)"gt"_(3)^(2) " "cdots(3)`
Adding first two expressions
`h((1)/(t_(1))+(1)/(t_(2)))=(1)/(2)g(t_(1)+t_(2))`
Comparing it with eqn. (3)
`t_(3) = sqrt(t_(1)t_(2))`