Question

The height and width of each step of a staircase are 20 cm and 30 cm respectively. A ball rolls off the top of a stair with horizontal velocity v and hits the fifth step. The magnitude of v is?
A. 22.42
B. 43.23
C. 12.12
D. 33.54

Answer 1

The x-axis represents the width of the staircase and the y-axis represents the height of the staircase. Using these statements in terms of the equation of motion, we will find the relation between the velocity and the dimensions of the staircase. Finally, substituting the values, we will compute the magnitude of the velocity.
Formula used:
$s=ut+\dfrac{1}{2}a{{t}^{2}}$

Complete answer:
From the given information, we have the data as follows.
The height of the staircase, $h=20\,cm$
The width of the staircase, $w=30\,cm$
The number of stairs, $n=5$
Let ‘b’ represent the width of the staircase and ‘h’ represents the height of the staircase. Let ‘u’ be the initial velocity of the ball, ‘t’ be the time taken by the ball to reach the ground, ‘g’ be the acceleration due to gravity.
The velocity of the ball concerning the x-axis is, ${{u}_{x}}=0$. The velocity of the ball concerning the y-axis is, ${{u}_{y}}=0$.
The acceleration of the ball concerning the x-axis is, ${{a}_{x}}=0$. The acceleration of the ball concerning the y-axis is, ${{a}_{y}}=g$.
Thus, the equations of x and y are given as follows.
$x=ut$and $y=0+\dfrac{1}{2}g{{t}^{2}}$
Represent the equation of ‘x’ in terms of ‘t’.
$t=\dfrac{x}{u}$
Substitute this equation of time in the equation of y.

& y=\dfrac{1}{2}g{{\left( \dfrac{x}{u} \right)}^{2}} \\\ & \therefore y=\dfrac{g{{x}^{2}}}{2{{u}^{2}}} \\\ \end{aligned}$$……. (a) The equations of x and y in terms of the number of staircases, the width and the height of the staircase is given as follows. $$x=nb$$and $$y=nh$$ Here ‘n’ represents the number of staircases, ‘b’ represents the width and ‘h’ represents the height of the staircase. Substitute these expressions in equation (a). $$\begin{aligned} & (nh)=\dfrac{g}{2{{u}^{2}}}{{\left( nb \right)}^{2}} \\\ & \Rightarrow h=\dfrac{gnb}{2{{u}^{2}}} \\\ & \therefore {{u}^{2}}=\dfrac{gnb}{2h} \\\ \end{aligned}$$ Thus, the expression for the magnitude of the velocity of the ball is given as follows. $$u=\sqrt{\dfrac{gnb}{2h}}$$ Substitute the given values in the above equation. $$\begin{aligned} & u=\sqrt{\dfrac{10\times 5\times {{30}^{2}}}{2\times 20}} \\\ & \Rightarrow u=\sqrt{1125} \\\ & \therefore u=33.54\, \\\ \end{aligned}$$ $$\therefore $$ The magnitude of v is 33.54. **Thus, option (D) is correct.** **Note:** The derivation of how we have arrived at the equations either for computing the number of staircases, the initial velocity, the width of the staircase and the height of the staircase is explained.