Source: Principles of Physics
Problem Sets:
Problem
A ball rolls horizontally off the top of a stairway with an initial speed $v_0$. The steps each have height $h$ and width $w$.
The ball first hits step $n$, where $n$ is the smallest integer satisfying the condition:
$$n > \frac{2v_0^2 h}{gw^2}$$Let the origin be the edge of the top step, with the x-axis horizontal and the y-axis vertically downward. The ball's position is given by the equations of projectile motion:
$$x(t) = v_0 t$$ $$y(t) = \frac{1}{2}gt^2$$By eliminating the time variable $t = x/v_0$, we find the ball's trajectory equation:
$$y(x) = \frac{g}{2v_0^2}x^2$$The top-front corner of the $n$-th step is located at the coordinates $(nw, nh)$. The ball will land on or beyond the $n$-th step if its trajectory passes below this corner. This means that at the horizontal position $x = nw$, the ball's vertical position $y$ must be greater than $nh$.
$$y(x=nw) > nh$$Substituting into the trajectory equation gives the condition:
$$\frac{g}{2v_0^2}(nw)^2 > nh$$Assuming $n>0$, we can simplify and solve for $n$:
$$\frac{gnw^2}{2v_0^2} > h$$ $$n > \frac{2v_0^2 h}{gw^2}$$The step number must be an integer. The first step the ball can possibly hit is the smallest integer $n$ that satisfies this inequality.