Question

Einstein's most famous equation is \(E=m c^{2}\). In this equation, \(E\) stands for energy, \(m\) stands for mass, and \(c\) stands for the speed of light. Use algebra to solve this equation for the mass. That is, complete this equation: $$ m=? $$

Short answer

\[ m = \frac{E}{c^2} \]

Step-by-step solution

Identify the Formula

The given equation is Einstein's famous formula for energy:\[ E = m c^2 \]Where \(E\) stands for energy, \(m\) stands for mass, and \(c\) stands for the speed of light.

Isolate the Variable

To solve for \(m\), we need to isolate it on one side of the equation. We'll do this by dividing both sides of the equation by \(c^2\). This removes \(c^2\) from the right side of the equation:\[ m = \frac{E}{c^2} \]

Simplify and Present the Solution

Now that \(m\) is isolated, the simplified solution for mass \(m\) in terms of energy \(E\) and the speed of light \(c\) is:\[ m = \frac{E}{c^2} \]This equation allows us to calculate the mass from a known energy and speed of light.

Key concepts

Mass-Energy Equivalence

Einstein's equation, \(E=mc^2\), is a fundamental principle of modern physics known as mass-energy equivalence. This concept signifies that mass and energy are two forms of the same thing, interconnected in such a way that they can be converted into one another. Before Einstein's discovery, energy and mass were considered completely separate entities. His equation demonstrated that a small amount of mass can be transformed into a significant amount of energy, exemplified by atomic reactions.
Imagine energy as the capacity to cause change or do work, and mass as a measure of an object's inertia or resistance to motion. In Einstein's equation, this relationship is represented as the product of mass \(m\) and the square of the speed of light \(c^2\).

• \(E\) represents energy, a measurable quantity that can be transferred between systems.
• \(m\) represents mass, the quantity of matter in an object.
• \(c\), the speed of light, is a constant that bridges these two concepts.

Einstein's mass-energy equivalence has practical implications, such as in nuclear energy and understanding particle physics.

Speed of Light

The speed of light, denoted as \(c\), is a universal constant and one of the fundamental constants of nature. It is valued at approximately \(299,792,458\) meters per second (or about \(3 \times 10^8\) m/s). This speed defines how fast light travels in a vacuum and serves as a crucial factor in Einstein's equation.
When you see \(c^2\) in the equation \(E=mc^2\), it emphasizes how energy is related to mass with the speed of light squared as a proportional factor. Because \(c\) is an extraordinarily large number, even a small amount of mass can equate to a large amount of energy.

• The speed of light is the ultimate speed limit in the universe; nothing can travel faster than light.
• It is a constant that applies universally, making \(E=mc^2\) the same everywhere in the universe, irrespective of location or conditions.

This constancy is vital in our understanding of physics and cosmology, helping us comprehend phenomena such as time dilation and redshift.

Algebraic Manipulation

Algebraic manipulation involves using algebra to solve equations or expressions, like isolating a variable. In the context of the equation \(E=mc^2\), solving for mass \(m\) involves a step-by-step algebraic process.
Here's how it works:
1. **Identify the Original Equation:** Start with the known equation \(E=mc^2\). This formula relates energy to mass and the speed of light.2. **Isolate the Desired Variable:** To find \(m\), isolate it by dividing both sides of the equation by \(c^2\), which cancels out \(c^2\) on the right side. We get \(m=\frac{E}{c^2}\).3. **Simplify the Expression:** Now, \(m\) is expressed in terms of \(E\) and \(c\), simplifying the equation as \(m=\frac{E}{c^2}\). This form shows that mass is equivalent to energy divided by the speed of light squared.
The process of algebraic manipulation is critical in physics for re-arranging formulas to solve for various quantities. It allows us to adapt equations to fit different scenarios in scientific problems, making it an essential tool for students and professionals alike.