A motor is used to deliver water at a certain | vedclass

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(C) Let the density of water be $\rho$,the cross-sectional area of the pipe be $A$,and the velocity of water be $v$.The mass of water flowing out per

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$n^3$-times

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(C) Let the density of water be $\rho$,the cross-sectional area of the pipe be $A$,and the velocity of water be $v$.The mass of water flowing out per second is given by $m = A v \rho$.The kinetic energy imparted to the water per second is the power $P$ required by the motor:$P = \frac{1}{2} m v^2 = \frac{1}{2} (A v \rho) v^2 = \frac{1}{2} A \rho v^3$.This implies that $P \propto v^3$.If we want to deliver $n$-times the mass of water in the same time,the mass flow rate $m' = n m$.Since $m = A v \rho$,we have $m' = A v' \rho = n (A v \rho)$,which implies $v' = n v$.The new power $P'$ required is $P' = \frac{1}{2} A \rho (v')^3$.Taking the ratio of the new power to the original power:$\frac{P'}{P} = \frac{\frac{1}{2} A \rho (n v)^3}{\frac{1}{2} A \rho v^3} = n^3$.Therefore,the power must be increased $n^3$-times.

$A$ motor is used to deliver water at a certain rate through a given horizontal pipe. To deliver $n$-times the water through the same pipe in the same time,the power of the motor must be increased as follows:

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