A 57 -kg woman runs up a flight of stairs having a rise of \(4.5 \mathrm{~m}\) in \(3.5 \mathrm{~s}\). What average power must she supply?

Understanding gravitational potential energy (GPE) is crucial when calculating the power required for tasks involving height changes, such as climbing stairs. GPE is the energy that an object possesses due to its position in a gravitational field. The formula to calculate it is:
\[ PE_{\text{gravity}} = m \times g \times h \]
Here,

• \( m \) represents the mass of the object (in kilograms),
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \) on Earth), and
• \( h \) is the height above the reference point (in meters).

In our textbook problem, a woman with a mass of 57 kg climbs a flight of stairs rising 4.5 m in 3.5 seconds. To find out how much work she does against gravity, we use her mass, the height of the stairs, and the gravitational acceleration to determine the increase in her GPE. By understanding the GPE concept, students can connect an object's elevated position to its potential to perform work.

The work-energy principle is a fundamental concept in physics that relates the work done on an object to its energy change. Work is considered to be done when a force causes a displacement. The principle states that the work done by all forces acting on an object equals the change in its kinetic energy.
In the context of our textbook scenario, when the woman runs up the stairs, the work she does against gravity leads to an increase in gravitational potential energy, representing the gained height. Using the equation for work (which is also the GPE gained):
\[ W = m \times g \times h \]
We've calculated the work done to be 2513.1 Joules. This information paves the way for understanding how power is computed, since power is defined as the rate at which work is done. Breaking down complex calculations into steps based on physics principles like this can greatly aid students in problem-solving and comprehension.

Solving problems in physics often involves a step-by-step approach, where each step builds upon the previously established concepts. A well-structured problem-solving strategy can include identifying the given information, determining the relevant physics principles, executing precise calculations, and interpreting the results.
For the stair climbing power calculation problem, we begin by identifying the known quantities: the woman's mass (57 kg), the flight of stairs' rise (4.5 m), and the time taken (3.5 s). Next, we apply the relevant concepts: gravitational potential energy and the work-energy principle to calculate work done. With this foundation, we then advance to calculating the average power, which involves dividing the total work done by the time it took to do that work using the formula:
\[ P = \frac{W}{t} \]
Through this process, students learn not only to make calculations but also to understand the underlying principles of physics that govern the problem at hand. Encouraging this comprehensive approach enhances their ability to tackle a wide array of physics problems effectively.