Inflatable mattress

Let the minimum thickness of the mattress be $h$ and the velocity when the hit the mattress be $v_h$ . By the conservation of energy, we have $\frac 12 m v_h^2 = mg (10-h)$ , where $m$ is the mass of the boy and $g$ is the acceleration due to gravity. Then we have $v_h^2 = 2g(10-h)$ .

After hitting the mattress, the initial velocity of the boy $v_h$ is to be decelerated by a $10-G$ force until $0$ when he reach the ground level. Then we have $v_h^2 = - 20gh$ and that:

$\begin{aligned} 2g(10-h) & = 20gh \\ 11 h & = 10 \\ \implies h & = \frac {10}{11} \ \rm m \approx \boxed{91} \ cm \end{aligned}$

$\red{\text{Discussion:}}$ . But I think the deceleration force through the mattress should be 9-G instead of 10-G, because if the deceleration force is 10-G + the gravitational force of 1-G, the poor boy is taking a total of 1-G which may kill him. If net 9-G deceleration is considered the answer is a perfect $100\ \rm cm$ .

During the first $\frac{10}{11}$ th part of his fall, the boy will accelerate constantly at 1 g (gravitationnal field). During the last $\frac{1}{11}$ th part of his fall, the boy will decelerate constantly at 10 g in the mattress (extreme case of non injury) and will reach the floor at zero speed. That's why the mattress must have a thickness of $\frac{H}{11}$ =0.91 m