Question

A water hose 2 cm in diameter is used to fill a 20 L bucket. If it takes 1 minute to fill the bucket, with what velocity (in $cm \ s^{-1}$) it leaves the hose?
A. 150
B. 70
C. 106
D. 100

Answer 1

In such types of questions, if we are able to get the rate of flow of water into the bucket, we can get the velocity of water in the hose easily. Equation of continuity is an important result of Bernoillui’s theorem. In order to maintain a continuum, the rate of water flowing through the hose and rate of water going into the bucket must be the same.

Formula used:
$\dfrac{dQ}{dt} = Av$, where A is the area of hose and v is the velocity of water flowing through the hose. $\dfrac{dQ}{dt}$ is the rate of flow of water.

Complete answer:
The term $\dfrac{dQ}{dt}$ represents the rate of flow of water. It means total volume of water flown per unit time. Hence we can calculate this term with respect to bucket. Given a bucket, total volume filled = 20L = $20 \times 10^{-3} m^{3}$ in total time of 1 minute i.e. 60 seconds.
Hence $\dfrac{dQ}{dt} = \dfrac{20\times 10^{-3}}{60} = \dfrac 13 \times 10^{-3} m^{3}s^{-1}$
Now, using equation of continuity, we know $\dfrac{dQ}{dt} = Av$
Hence $Av =\dfrac{1}{3}\times 10^{-3}$
Now, given the diameter = 2 cm
Hence radius = 1cm = 0.01m
Thus, area = $\pi 0.01^2 = \dfrac \pi{10000}$
Hence, $v= \dfrac {\dfrac{1}{3}\times 10^{-3}}{\dfrac \pi {10000}} = 1.06 ms^{-1} = 106 \ cm \ s^{-1}$

So, the correct answer is “Option C”.

Note:
One could say that the equation of continuity means $A_1v_1=A_2v_2$. That is correct but the product of the area of pipe and velocity of liquid flowing in it is also called the rate of flow of liquid. Hence one must not confuse with it. Also students must focus on the units also. The conversion of litre to $m^3$is done so as to get the required result in S.I units.