Calculating the Power Required for a 30kg Boy to Run Up a Staircase

In this article, we will explore the scientific principles behind the power required for a boy of 30kg to run up a staircase of 40 steps in 10 seconds. We will break down the problem, apply the necessary formulae, and step through the calculations to ensure a thorough understanding of the concept.

Introduction

The problem at hand involves determining the amount of power required by a boy of mass 30kg to run up a staircase of 40 steps in 10 seconds. To solve this, we need to consider the acceleration due to gravity and the number of steps as well as the work done and the time taken.

Understanding the Problem

The key variables provided in the problem are:

Mass of the boy, ( m 30 ) kg Acceleration due to gravity, ( g 10 ) m/s2 Number of steps, ( n 40 ) Total time taken, ( t 10 ) seconds

However, in the problem, the height of each step is not clearly mentioned. For simplicity, let's assume the height of each step is 0.3 meters. This assumption will be used to calculate the height of the staircase and then the required power.

Step-by-Step Calculation

Step 1: Calculate the Total Height of the Staircase

Since each step has a height of 0.3 meters and there are 40 steps, the total height ( H ) of the staircase is:

[begin{aligned}text{Height of one step} 0.3 , text{meters} text{Total height} n times text{height of one step} H 40 times 0.3 H 12 , text{meters}end{aligned}]

Step 2: Calculate the Work Done

The work done by the boy to climb the staircase can be calculated as the product of the boy's mass, the acceleration due to gravity, and the total height climbed. The formula for work done is:

[begin{aligned}text{Work Done} text{mass} times text{acceleration due to gravity} times text{height} W m times g times H W 30 , text{kg} times 10 , text{m/s}^2 times 12 , text{m} W 3600 , text{Joules} end{aligned}]

Step 3: Calculate the Power Required

Power is defined as the work done per unit time. Therefore, the power required for the boy to climb the staircase in 10 seconds is given by:

[begin{aligned}text{Power} frac{text{Work Done}}{text{Time}} P frac{3600 , text{Joules}}{10 , text{seconds}} P 360 , text{Watts} end{aligned}]

Conclusion

The power required for a 30kg boy to run up a staircase of 40 steps in 10 seconds, assuming each step has a height of 0.3 meters, is 360 Watts. This calculation takes into account the application of gravitational potential energy, which is the work done against gravity to lift the boy's mass to a certain height.

By understanding the relationship between mass, acceleration due to gravity, height, work done, and power, we can better comprehend the physical effort and energy consumption involved in such tasks.