Equation of Continuity — Prelims Questions

Water flows through a horizontal pipe of varying cross-section. At a point where the diameter is $4,\text{cm}$, the velocity of water is $2,\text{m/s}$. What will be the velocity of water at another point where the diameter is $2,\text{cm}$?

An incompressible fluid flows through a pipe. At point P, the cross-sectional area is $0.5,\text{m}^2$ and the fluid velocity is $10,\text{m/s}$. If the fluid then flows through point Q where the velocity is $25,\text{m/s}$, what is the cross-sectional area at point Q?

A main pipe of radius $R$ carries water at a speed $V$. It then branches into $N$ smaller pipes, each of radius $r$. If the speed of water in each small pipe is $v$, which of the following expressions correctly relates these quantities?

Which of the following conditions is NOT assumed for the simplified Equation of Continuity ($A_1v_1 = A_2v_2$)?

A fluid flows through a pipe with a circular cross-section. If the radius of the pipe is halved, how does the fluid velocity change, assuming incompressible, steady flow?

A cylindrical tank of cross-sectional area $A_T$ is filled with water. A small hole of area $A_H$ is made at its bottom. If $A_H ll A_T$, and the water level in the tank is $h$, what is the velocity of efflux of water from the hole? (Assume ideal fluid and $g$ is acceleration due to gravity.)