(a) Calculate the energy released by the destruction of \(1.00 \mathrm{kg}\) of mass. (b) How many kilograms could be lifted to a \(10.0 \mathrm{km}\) height by this amount of energy?

One of the revolutionary concepts that came from Albert Einstein's theory of relativity is encapsulated in the famous equation: \(E = mc^2\). This expresses the principle of mass-energy equivalence, highlighting that mass and energy are two forms of the same thing and can be converted into each other.

In the equation, \(E\) represents energy, \(m\) symbolizes mass, and \(c\) is the speed of light in a vacuum, which is roughly \(3.00 \times 10^8 m/s\). Since the speed of light squared is an incredibly large number, even a small amount of mass can be converted into a huge amount of energy. This concept is not just theoretical; it is the foundation of various modern applications, including nuclear power and medical treatments using radiation.

Understanding this equation can help students not only solve textbook problems but also grasp the immense potential energy that lies within even the tiniest particles that make up our universe.

Potential energy (PE) refers to the energy stored in an object due to its position or configuration. The gravitational potential energy is particularly relevant to many physics problems and can be calculated using the formula: \(PE = mgh\).

Here, \(m\) stands for the mass of the object, \(g\) is the acceleration due to gravity on Earth which is approximately \(9.81 m/s^2\), and \(h\) represents the height above a reference point. When an object is lifted, work is done against the gravitational force, and this work becomes stored as potential energy.

For example, in our textbook problem, we are calculating how the energy released from mass can lift an object to a considerable height. This demonstrates not only how potential energy can be calculated but also how it can be transformed from one form of energy to another, adhering to the conservation of energy principle.

The principle of conservation of energy is a fundamental concept in physics that states that energy cannot be created or destroyed, only transformed from one form to another or transferred from one object to another. This means that the total energy in a closed system remains constant.

In context with our textbook problem, the energy (\(E\)) calculated from the mass-energy equivalence is not lost but instead is converted into potential energy which enables mass to be lifted. The ability to lift mass to a particular height is directly related to this amount of energy.

When solving physics problems, understanding energy conservation helps students predict how energy will flow or transform throughout the system. Whether it is potential energy being converted to kinetic energy or vice versa, this principle ensures that all energy changes are accounted for within a system.