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?
+26387 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.
+26387 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.
