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

Given:
Height of one step = 10 cm
Width of one step = 20 cm
Total height of the staircase = y = 30 cm
Total width of the staircase = x = 40 cm


To directly hit the lowest plane, the ball should just touch point E.
Let point A be the origin of reference coordinate.
Let u be the minimum speed of the ball.
We have:
x = 40 cm
y = −20 cm
θ = 0°
g = 10 m/s² = 1000 cm/s²
$\therefore y=x\tan \theta -g\frac{x^2{\sec }^2\theta }{2u^2}$
$$\begin{gathered} \Rightarrow -20=-\frac{800000}{2u^2} \\ \Rightarrow u=200\mathrm{cm}/\mathrm{s}=2\mathrm{m}/\mathrm{s} \end{gathered}$$
Thus, the minimum horizontal velocity of the ball is 2 m/s.