A pump is used to deliver water at a certain rate from a given pipe. To obtain $n$ times water from the same pipe in the same time, by what factor, the force of the motor should be increased?

Amount of water flowing per unit time $\frac{dm}{dt}=Av\rho$

$v=$velocity of flow, $A$ is area of cross-section

$\rho =$density of liquid

To get $n$ times water in the same time,

$$\begin{gathered} {\left(\frac{dm}{dt}\right)}^{\text{'}}=n\left(\frac{dm}{dt}\right) \\ \Rightarrow A v\text{'}\rho =nAv\rho \Rightarrow v\text{'}=nv \\ \Rightarrow F=v\left(\frac{dm}{dt}\right) \\ \Rightarrow F\text{'}=v\text{'}{\left(\frac{dm}{dt}\right)}^{\text{'}}=n^{2}v\left(\frac{dm}{dt}\right)=n^{2}F \end{gathered}$$

To get $n$ times water, force must be increased $n^2$ times.