Question

A motor pump of power 1 H.P is discharging water with a velocity of $10m{s^{ - 1}}$. In order to discharge water with a velocity of $20m{s^{ - 1}}$, the power of the motor should be
A. 2H.P.
B. 4H.P.
C. 8H.P.
D. 16H.P.

Answer 1

Power is the rate of doing work. The work-energy principle should be applied in this case. The power in each case for each value of velocity should be calculated and then the ratio between the two gives us the answer.

Formula Used: The formulae used in the solution are given here.
$P = \dfrac{W}{t}$ where $W$ is the work done, $t$ is the time taken and $P$ is the power.
The kinetic energy is given by, $K.E. = \dfrac{1}{2}m{v^2}$, when velocity is $v$ and mass is $m$.

Complete Step by Step Solution: It has been given that a motor pump of power 1 H.P is discharging water with a velocity of $10m{s^{ - 1}}$.
Now, we know that, $1H.P. \simeq 746W$.
We know that power is defined as the rate of doing work. Thus, mathematically, $P = \dfrac{W}{t}$ where $W$ is the work done, $t$ is the time taken and $P$ is the power.
Work done is the measure of energy transfer that occurs when an object is moved over a distance by an external force. The work-energy theorem states that the net work done by the forces on an object equals the change in its kinetic energy.
The kinetic energy is given by, $K.E. = \dfrac{1}{2}m{v^2}$, when velocity is $v$ and mass is $m$.
Therefore, $P = \dfrac{{\left( {\dfrac{1}{2}m{v^2}} \right)}}{t}$ .
Given that, $v = 10m{s^{ - 1}}$. Substituting,
$P = \dfrac{{\left( {\dfrac{1}{2}m \times {{10}^2}} \right)}}{t}$
Simplifying,
$P = \dfrac{{50m}}{t}$.
When $v = 20m{s^{ - 1}}$, the power is,
$P' = \dfrac{{\left( {\dfrac{1}{2}m \times {{20}^2}} \right)}}{t}$
Simplifying,
$P' = \dfrac{{200m}}{t}$
When, $P = \dfrac{{50m}}{t} = 1H.P.$
Then, $P' = \dfrac{{200m}}{t} = 4(\dfrac{{50m}}{t}) = 4H.P.$.
Hence, the correct answer is Option B.

Note: Horsepower (H.P.) is a unit of measurement of power, or the rate at which work is done, usually in reference to the output of engines or motors. There are many different standards and types of horsepower. Two common definitions used today are the mechanical horsepower (or imperial horsepower), which is about 746 watts.