Question

A staircase contains three steps each 10 cm high and 20 cm wide (figure 3−E9). What should be the minimum horizontal velocity of a ball rolling of the uppermost plane so as to hit directly the lowest plane?
Figure

Answer 1

The minimum horizontal velocity of a ball rolling of the uppermost plane so as to hit directly the lowest plane is 2 m/s.

Explanation:

Step 1:

To hit directly the lowest plane the ball has to cross the point B.

From A to B, the  ball has to travel to horizontal distance equal 40 cm =0.4 m

and vertical distance of 20 cm=0.2 m.

Step 2:

Let the time taken to reach point be t.

u=0 m/s

h=0.2 m

from the equation,

$u=u t+\frac{1}{2} g t^{2}$

we get

$\begin{aligned}&0.2=0+\frac{1}{2} 9.8 t^{2}\\&t^{2}=\frac{0.2 \times 2}{9.8}\end{aligned}$

$\begin{aligned}&t^{2}=\frac{0.4}{9.8}\\&t^{2}=0.04\end{aligned}$

$t^{2}=0.2$

Step 3:

Let v be the horizontal velocity of ball than the horizontal distance travelled by the ball = 0.2 v

in order to clear the point B it must be equal to 0.4

0.2 v=  0.4

$\begin{aligned}&v=\frac{0.4}{0.2}\\&v=2 m / s\end{aligned}$

∴So minimum velocity required to to hit the lowest floor  is 2 m/s.