Question

A particle is projected with some speed in vertically upward direction it hits the ground after t1 second it is projected downward with same speed it hits the ground after t2 seconds find time taken by particle to hit the ground if it is dropped from same height

Answer 1

Just apply second equation of motion in three cases and use algebraic manipulation.
$0=ut_1-\dfrac{1}{2}g{t_1}^2\\\Rightarrow u=\dfrac{gt_1}{2}\qquad(1)\\\text{Also,}\\h=ut_2+\dfrac{1}{2}g{t_2}^2\qquad(2)\\\text{Now,}\\h=\dfrac{1}{2}gt^2\\\text{Plug in }(2),\\\Rightarrow ut_2+\dfrac{1}{2}g{t_2}^2=\dfrac{1}{2}gt^2\\\text{Plug in }(1),\\\Rightarrow \dfrac{1}{2}gt_1t_2+\dfrac{1}{2}g{t_2}^2=\dfrac{1}{2}gt^2\\t=\sqrt{t_2(t_1+t_2)}$