Question

Two men with weights in the ratio $4:3$ run up a staircase in time in the ratio $12:11$. The ratio of power of the first to that of second is:
A. $\dfrac{4}{3}$
B. $\dfrac{{12}}{{11}}$
C. $\dfrac{{48}}{{33}}$
D. $\dfrac{{11}}{9}$

Answer 1

The above problem can be resolved using the concepts and mathematical expressions for power in kinematics. The mechanical power in kinematics is expressed by dividing the work done and the time taken to do that work. In this problem, we are given with the ratio of weights and time required to climb the stairs for the respective individual. We can apply the modified expression for the power by involving the weight, height and time. And in this, the height will remain the same for both men, as they need to climb the same distance.

Complete step by step answer:
Given:
The ratio of weights is, ${W_1}:{W_2} = 4:3$.
The ratio of time is, ${t_1}:{t_2} = 12:11$.
We know that the expression for the power is,

$$\Rightarrow P = \dfrac{{weight \times height}}{{time}}\\\ \Rightarrow P = \dfrac{{W \times h}}{t}$$

Let us consider the subscript 1 to denote the man 1 and subscript 2 to denote the man 2.
As, the height is the same for both. Then take the ratio of power as,

$$\Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{\left( {\dfrac{{{W_1} \times h}}{{{t_1}}}} \right)}}{{\left( {\dfrac{{{W_2} \times h}}{{{t_2}}}} \right)}}\\\ \Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{{W_1}}}{{{W_2}}} \times \dfrac{{{t_2}}}{{{t_1}}}$$

Solve by substituting the numerical values in above expression as,

$$\Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{{W_1}}}{{{W_2}}} \times \dfrac{{{t_2}}}{{{t_1}}}\\\ \Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{4}{3} \times \dfrac{{11}}{{12}}\\\ \Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{44}}{{36}}\\\ \Rightarrow \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{11}}{9}$$

Therefore, the ratio of power of the first to that of second is $\dfrac{{11}}{9}$ andoption (D) is correct.

Note: To resolve the above problem, one must know the concept and fundamentals involved in the calculation of work and power. There is a basic relation between the power and the work done, as the power is calculated by dividing the magnitude of work done and the time required to finish the work. Moreover, it is also important to remember several modified forms of work and power, to be handy with the problem resolving tools.