Question

The power of a scooter is rated at $5HP$. In practice, it attains a speed of $54kmph$ in $2s$and it weighs $100kg$. What is the gain or loss of power?
A. Gain $1HP$
B. Gain $2.54HP$
C. Loss $1HP$
D. Loss $25HP$

Answer 1

In this problem, to determine the gain or loss of power, we need to find the kinetic energy of the scooter during the given time. For that, first we will convert the given speed in SI units. Then we will find the kinetic energy using mass and velocity of the scooter. After that using the relation between power, energy and time, we will find the power and finally determine the difference of obtained power and rated power to find out the required gain or loss.

Formulas used:
$K.E. = \dfrac{1}{2}m{v^2}$, where, $K.E.$is the kinetic energy of the scooter, $m$is the mass of the scooter, $v$ is the speed of the scooter
$P = \dfrac{{K.E.}}{t}$, where, $P$is the power, $K.E.$is the kinetic energy of the scooter and $t$ is the time

Complete step by step answer:
The speed of the scooter is given as $54kmph$. We will first convert it into $m/s$.
$v = 54kmph = \dfrac{{54 \times 1000}}{{3600}} = 15m/s$
Now, we will find the kinetic energy of the scooter.
$K.E. = \dfrac{1}{2}m{v^2} \\\ \Rightarrow K.E. = \dfrac{1}{2} \times 100 \times {15^2} = 11250J \\\$
We know that Power is given by
$P = \dfrac{{K.E.}}{t} \\\ \Rightarrow P = \dfrac{{11250}}{2} \\\ \Rightarrow P= 5625\dfrac{J}{s} \\\$
The obtained power is greater than the rated power and hence there is a gain in power.
$\therefore$ Gain in power $= 7.54 - 5 = 2.54HP$

Hence, option B is the right answer.

Note: While solving this type of question, we need to be careful in unit conversation, otherwise the answer will be wrong. Here, it is very important to convert speed into SI units and then start the calculation. Besides this, at the end, the conversion of power from joule per second to horsepower is very important as we need to find the power difference in the horsepower unit.