A supemova explosion of a \(2.00 \times 10^{31} \mathrm{kg}\) star produces \(1.00 \times 10^{44} \mathrm{J} \quad\) of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio \(\Delta m / m\) of mass destroyed to the original mass of the star?

Imagine witnessing an astronomical event so powerful that it momentarily outshines an entire galaxy. This is what happens during a supernova explosion—a colossal explosion that marks the death of a massive star.

When a star runs out of nuclear fuel, it can no longer support itself against the pull of gravity. For very massive stars, this leads to the core collapsing and rebounding, creating a shockwave that propels the outer layers into space. This violent expulsion releases an unbelievable amount of energy into the cosmos. To put this into perspective, the energy output from a supernova can match the energy the Sun is expected to emit over its entire 10-billion-year lifetime—but a supernova releases this in mere seconds or minutes.

Supernovae play a critical role in the universe, not only in dispersing elements across star systems but also in catalyzing the formation of new stars. Without supernovae, many of the elements essential for life as we know it, wouldn't exist.

One of the most celebrated equations in physics is Einstein's mass-energy equivalence formula, written as E=mc^2. This equation describes the relationship between mass (m) and energy (E), with c representing the speed of light in a vacuum, approximately 299,792,458 meters per second.

This elegant and deceptively simple formula implies that mass can be converted into energy and vice versa. The 'c^2' in the equation tells us that even a small amount of mass can be transformed into a disproportionately large amount of energy, due to the speed of light being such a large number. This conversion principle is not merely theoretical; it's practically demonstrated in nuclear reactions and matter-antimatter annihilation, where mass is destroyed to release vast quantities of energy.

Special relativity is a theory of time and space formulated by Albert Einstein in his groundbreaking paper of 1905. It is based on two postulates: firstly, the laws of physics are the same in all inertial frames of reference, and secondly, the speed of light in a vacuum is constant, regardless of the motion of the light source or observer.

One of the most counterintuitive consequences of special relativity is time dilation, where time can pass at different rates for observers moving relative to one another. Special relativity also led to the understanding that space and time are interwoven into a single continuum—spacetime. It changed our perception of the universe and set the stage for the development of the mass-energy equivalence principle, encapsulated in the equation E=mc^2.

The principle of conservation of mass-energy is derived from the laws of thermodynamics and Einstein's theory of special relativity, and it states that within a closed system, mass and energy can neither be created nor destroyed; they can only be transformed from one to another.

In the context of a supernova, when a star explodes, it does not lose mass; rather, a portion of its mass is converted into energy, adhering to the conservation law. This transformation is a powerful demonstration of E=mc^2 in action. During such an event, an enormous quantity of energy is released, affecting the surrounding space and contributing to the evolution of the galaxy and the universe.

Understanding conservation of mass-energy enables scientists to predict the behavior of different physical systems, from the inner workings of the atom to the grand scale events like supernova explosions. It is a fundamental concept that underpins much of modern physics, reminding us that everything in the universe is interconnected and nothing gets truly lost, just changed.