Question

A boy of mass $50kg$ runs up a staircase of $45$ steps in $9s$ . If the height of each step of the staircase is $15cm$ , find the power of the boy.
A. $375W$
B. $480W$
C. $550W$
D. $5390W$

Answer 1

Hint: To calculate the power of the boy we need to calculate his energy spent per unit time. The energy spent by the boy will be nothing but the difference in his gravitational potential energy at the bottom and the top of the staircase. He expends his own muscular energy to gain potential energy.
Formula used:
$\text{Power = }\dfrac{\text{Total energy spent }}{\text{Total time taken}}$
The change in gravitational potential energy $PE$ of a body of mass $m$ when it attains a height change $\Delta h$ is given by
$PE=mg\Delta h$
where $g$ is the acceleration due to gravity.

Complete step by step answer:
Given that mass of the body, $m=50kg$
Height of each step $=15cm$
Total height $h$ of 45 steps $45\times 15=675cm=6.75m$ $\left( \because 1cm=0.01m \right)$

Time $t=9s$
We take acceleration due to gravity
$g=9.8m/{{s}^{2}}$
Work needs to be done in lifting an object to a height=potential energy at the height
The energy spent by the boy is equal to the potential energy of the boy after climbing 45 steps.
The energy possessed by a body due to its position or configuration is called potential energy.

The change in gravitational potential energy $PE$ of a body of mass $m$ when it attains a height change $\Delta h$ is given by
$PE=mg\Delta h$ --(1)
where $g$ is the acceleration due to gravity.
Hence, using (1), we get the change in potential energy $\Delta PE$ of the boy will be
$\Delta PE=mgh$
$\therefore \Delta PE=50\times 9.8\times 6.75=3307.5J$ --(2)

The same amount of energy is spent by the boy to attain this increase in potential energy.
Now,
$\text{Power = }\dfrac{\text{Total energy spent }}{\text{Total time taken}}$ --(3)
The time taken by the boy to climb the staircase is given to be $t=9s$.
Hence, using (2) in (3), we get the power $P$ of the boy as
$P=\dfrac{3307.5}{9}=367.5W$
The option closest to this is A) $375W$, which would have been the answer if we had considered $g=10m/{{s}^{2}}$.

Note: Students should know that the change in the gravitational potential energy only depends upon the change in the height of the body’s position. It does not depend on the path taken by the body to attain the final change. Hence, a body can go in multiple steps to reach the final height or even go the final height in one step and its change in gravitational potential energy would be the same. This is because the force of gravity is a conservative force which means that the work done by it only depends upon the initial and final positions of the body and not the path taken by the body. This concept is often used to make confusing questions to trick students, especially in competitive exams.