A pump motor is used to deliver water at a certain rate from a given pipe. To obtain twice as much water from the same pipe in the same time, power of the motor has to be increased to

The correct option is 2 times.

To understand why the power of the motor must be increased to 2 times, let us analyze the relation between the power of a pump motor, the work done, and the mass of the water delivered.

Step 1: Understanding Power and Work Done
The power of a motor ($P$) is defined as the rate at which work ($W$) is done over a given time interval ($t$):
$P=\frac{W}{t}$

Step 2: Relate Work Done to the Mass of Water
When lifting or delivering water of mass $m$ to a height $h$, the work done by the pump is equal to the potential energy gained by the water:
$W=mgh$
Here, $g$ represents the acceleration due to gravity.

Step 3: Formulating the Initial Power
Substituting the work equation into the power formula gives:
$P=\frac{mgh}{t}$

Step 4: Finding the New Power Required
We want to obtain twice as much water in the same time interval. This means the new mass of water $m^{\prime }$ is:
$m^{\prime }=2m$
Substituting this new mass into our power equation for the new power $P^{\prime }$, keeping height $h$, gravity $g$, and time $t$ constant, we get:
$P^{\prime }=\frac{m^{\prime }gh}{t}=\frac{2mgh}{t}=2P$

Therefore, the power of the motor has to be increased to 2 times the original power.