Answer
The marble strikes the fourth step first.
Work Step by Step
Since the horizontal velocity is $3.0~m/s$, and the width of each step is $0.30~m$, the marble can move horizontally past one step each 0.1 seconds. Therefore, the marble can pass $n$ steps horizontally in a time of $(0.1~n)~seconds$
We can find the vertical displacement the marble falls in a time of $(0.1~n)$ seconds:
$\Delta y = \frac{1}{2}at^2$
$\Delta y = \frac{1}{2}(9.8)(0.1~n)^2$
$\Delta y = (0.049~n^2)~meters$
We can find the total height of $n$ steps:
$h = (0.18~n)~meters$
We need to find the lowest integer $n$ such that $\Delta y \gt h$:
$(0.049~n^2) \gt (0.18~n)$
$(0.049~n) \gt (0.18)$
$n \gt \frac{0.18}{0.049}$
$n \gt 3.67$
$n = 4$ is the lowest integer such that $\Delta y \gt h$. Therefore, the marble strikes the fourth step first.
