A ball rolls on the top of a stairway with a horizontal velocity of 2.0 m/s. The steps are each 20.0 cm high and 20.0 cm wide. Which step wi

A ball rolls on the top of a stairway with a horizontal velocity of 2.0 m/s.

The steps are each 20.0 cm high and 20.0 cm wide.

Which step will the ball hit first ?

$$\\\small{\text{horizontal move: $\boxed{ x=v_0\cdot t} \; \Rightarrow \; t =\dfrac{x}{v_0} \qquad v_0=2\ \frac{m}{s}$ }}\\\\ \small{\text{vertical move: $\boxed{ y=\frac{g}{2}\cdot t^2 } \quad t =\dfrac{x}{v_0} \; \Rightarrow \; y(x)=\dfrac{g}{2}\cdot \dfrac{x^2}{v_0^2} \qquad g=10\ \frac{m}{s^2}$ }}\\\\ \small{\text{$x=y(x) ?:\qquad x=\dfrac{g}{2}\cdot \dfrac{x^2}{v_0^2} \; \Rightarrow \; x=\dfrac{2\cdot v_0^2 }{g} $ }}\\ \small{\text{ $ x=\dfrac{2\cdot (2\cdot \frac{m}{s})^2 }{10\cdot \frac{m}{s^2}} \qquad x=\frac{8}{10}\cdot m = 0.8\ m \qquad 0.8\ m = 4 * 0.2\ m $ }}$$

The ball will hit first step 4.