Consider the configuration of a stationary wa | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The velocity of efflux from the hole is given by $v_{e} = \sqrt{2gh}$.The horizontal range $x$ of the water jet is given by $x = v_{e} 	imes t$,w

Vedclass · Accepted Answer

$0.1$

Vedclass · Answer

(C) The velocity of efflux from the hole is given by $v_{e} = \sqrt{2gh}$.The horizontal range $x$ of the water jet is given by $x = v_{e} 	imes t$,where $t$ is the time of fall from height $H$.Since $H = \frac{1}{2}gt^{2}$,we have $t = \sqrt{\frac{2H}{g}}$.Thus,$x = \sqrt{2gh} 	imes \sqrt{\frac{2H}{g}} = 2\sqrt{hH}$.The speed $v$ of the bucket must match the rate of change of the range $x$ to keep the water falling into it:$v = \frac{dx}{dt} = \frac{d}{dt}(2\sqrt{hH}) = 2\sqrt{H} \frac{d}{dt}(\sqrt{h}) = 2\sqrt{H} \cdot \frac{1}{2\sqrt{h}} \cdot \frac{dh}{dt} = \sqrt{\frac{H}{h}} \cdot \frac{dh}{dt}$.From the equation of continuity,$A_{0} \left(-\frac{dh}{dt}ight) = A v_{e} = A \sqrt{2gh}$.Therefore,$\frac{dh}{dt} = -\frac{A}{A_{0}} \sqrt{2gh}$.Substituting this into the expression for $v$ (taking magnitude): $v = \sqrt{\frac{H}{h}} \cdot \frac{A}{A_{0}} \sqrt{2gh} = \frac{A}{A_{0}} \sqrt{2gH}$.Given $A = 5 \, cm^{2}$,$A_{0} = 500 \, cm^{2}$,$g = 10 \, m/s^{2}$,and $H = 5 \, m$:$v = \left(\frac{5}{500}ight) \sqrt{2 	imes 10 	imes 5} = \frac{1}{100} 	imes \sqrt{100} = \frac{10}{100} = 0.1 \, m/s$.

Similar Questions

$A$ tank of height $5\, m$ is full of water. There is a hole of cross-sectional area $1\, cm^2$ in its bottom. The initial volume of water that will come out from this hole per second is

There is a hole at the bottom of a water tank. The total pressure at the bottom is $3 \, atm$ $(1 \, atm = 10^{5} \, N/m^{2})$. What is the velocity of the water coming out of the hole?

$A$ large tank filled with water to a height $h$ is to be emptied through a small hole at the bottom. The ratio of times taken for the level of water to fall from $h$ to $h/2$ and from $h/2$ to zero is

The water is filled up to a height of $12\, \text{m}$ in a tank having vertical sidewalls. $A$ hole is made in one of the walls at a depth $h$ below the water level. The value of $h$ for which the emerging stream of water strikes the ground at the maximum range is $.....\, \text{m}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module