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(B) The total time of flight for the particle is $T = t_1 + t_2$.Since the particle is projected from the ground and returns to the ground,the time ta

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$\frac{1}{2} g t_1 t_2$

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(B) The total time of flight for the particle is $T = t_1 + t_2$.Since the particle is projected from the ground and returns to the ground,the time taken to reach the maximum height is $t_{max} = \frac{T}{2} = \frac{t_1 + t_2}{2}$.Using the equation of motion $v = u - gt$,at the maximum height $v = 0$,so $0 = u - g(\frac{t_1 + t_2}{2})$,which gives the initial velocity $u = \frac{g(t_1 + t_2)}{2}$.The height $H_B$ of point $B$ at time $t_1$ is given by $H_B = ut_1 - \frac{1}{2}gt_1^2$.Substituting $u$,we get $H_B = \frac{g(t_1 + t_2)}{2} t_1 - \frac{1}{2}gt_1^2$.$H_B = \frac{gt_1^2 + gt_1t_2 - gt_1^2}{2} = \frac{1}{2} g t_1 t_2$.

$A$ particle is projected vertically upwards from a point $A$ on the ground. It takes $t_1$ time to reach a point $B$ but it still continues to move up. If it takes further $t_2$ time to reach the ground from point $B$,then the height of point $B$ from the ground is:

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