Find the potential energy of a 70 -kg hiker (a) atop New Hampshire's Mount Washington, \(1900 \mathrm{m}\) above sea level, and (b) in Death Valley, California, 86 m below sea level. Take the zero of potential energy at sea level.

The concept of gravitational potential energy (GPE) is an essential part of physics involving an object's position relative to a gravitating body, like Earth. It reflects the potential an object has to do work as a result of gravity pulling it downwards. The formula to calculate GPE is \(PE = m \times g \times h\), where \(m\) represents the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) denotes the height above or below a reference point.

In educational exercises, demonstrating the calculation of GPE helps students understand how different factors affect an object's stored energy. When solving such problems, it is vital to consider the reference point, or the 'zero' level, from which height is measured. Our example considered sea level as the zero point. Thus, a hiker atop a mountain or below sea level has a measurable GPE because of their height relative to sea level. The key takeaway is that GPE can be easily calculated with a simple formula as long as the mass, gravity, and height are known.

In physics, the acceleration due to gravity, represented by the symbol \(g\), is the acceleration that the Earth imparts to objects on or near its surface. Its standard value is roughly \(9.8 \, \text{m/s}^2\), but it can slightly vary depending on altitude and geographical location. This uniform acceleration is what pulls objects towards the planet’s center.

Understanding \(g\) is crucial when calculating gravitational potential energy. It acts as a constant in the formula, attributed to Earth's gravitational force. However, for students to fully grasp this concept, it's essential to clarify that \(g\) is the reason why all objects, regardless of mass, fall at the same rate in the absence of air resistance. This uniformity allows us to simplify many physics problems by using a standard value for \(g\), thereby ensuring consistency across various calculations involving potential energy.

The height in potential energy equations symbolizes how far an object is from the reference point, which influences its potential energy in a gravitational field. In our example, the hiker's height above or below sea level directly impacts the computed GPE. Height \(h\) in the formula \(PE = m \times g \times h\) is taken as positive when the object is above the designated zero level and negative when below.

When height changes, it directly affects the potential energy value. For instance, if the hiker climbs higher, their potential energy would increase proportionally because they have a greater distance to fall under gravity's influence. Similarly, descending below the reference level would normally result in a negative GPE, which conceptually signifies that you would need to supply energy to bring the object back up to the zero level. However, in our conventional approach, we often disregard negative values for potential energy, simply using them to indicate a position below the reference point rather than a true negative energy. This concept aids students in conceptualizing how the change in an object's elevation alters its stored energy in the field of Earth's gravity.