Falling in Steps

A block is initially positioned at height $y = 1 \, \text{m}$ above the ground. There are infinitely many ladder rungs below the block.

Rung $1$ is a distance of $\frac{1}{2} \, \text{m}$ below the block's starting position. Rung $2$ is a distance of $\frac{1}{4} \, \text{m}$ below Rung $1$ . Rung $3$ is a distance of $\frac{1}{8} \, \text{m}$ below Rung $2$ , etc, etc.

Each time the block hits a rung, it comes to a stop and immediately begins falling again under the gravitational acceleration $g = 10 \, \text{m/s}^2$ . The block initially starts falling from its starting position $(y = 1 \, \text{m})$ at time $t = 0$ .

At what time does the block reach the ground?

Note: Assume that the block's size is negligible