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Chapter 2: Problem 22

Solve each formula for the indicated letter. \(E=m c^{2},\) for \(c^{2}\)

Short Answer

Expert verified
\(c^2 = \frac{E}{m}\)

Step by step solution

01

- Identify the given formula

The given formula is Einstein's mass-energy equivalence formula: \(E = m c^2\)
02

- Isolate the indicated variable

The goal is to solve for \(c^2\). To do this, we need to isolate \(c^2\) on one side of the equation. Start by dividing both sides of the equation by \(m\): \[ \frac{E}{m} = c^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving algebraic equations, it's often necessary to isolate a specific variable. This means manipulating the equation to get the desired variable alone on one side of the equation.
Here, we start with Einstein's formula, which is:
To solve for a specific variable, perform operations that will leave the targeted variable by itself.
Steps to Isolate a Variable:
  • Identify the variable you need to isolate.
  • Use inverse operations to move other terms to the opposite side of the equation.
  • Make sure to perform the same operation on both sides to maintain equality.
In our specific problem, we isolate
Starting with the equation:


We want to solve for . To get c^2 alone, we need to move the term m that's multiplied with c^2 to the other side of the equation. We achieve this by dividing both sides by m:
Inverse operations are essential in isolating variables as they help eliminate unwanted terms.
By following this method, we simplified the equation and solved for the desired variable.

Mass-Energy Equivalence
The concept of mass-energy equivalence is one of the cornerstone principles of physics and is described by Einstein's famous equation
It shows that mass can be converted into energy and vice versa.
Universal Applications to Mass-Energy Equivalence:
  • Nuclear fusion in the sun converts hydrogen atoms' mass into energy, powering solar radiation.
  • Nuclear fission in reactors transforms the mass of atoms like uranium into large amounts of energy.
The idea is that even small amounts of mass can be turned into significant energy.
Implications of Mass-Energy Equivalence:
  • Understanding the energy produced in star processes.
  • Developing nuclear energy technologies.
  • Exploring possibilities in theoretical physics and advanced energy forms.

This principle highlights the deep relationship between mass and energy, fundamentally altering our understanding of how the universe works.
Einstein's Formula
Einstein's formula, is fundamental to the theory of relativity. It provides a way to understand how mass and energy interact. Understanding Einstein's Formula: represents the energy of an object, represents the mass of the object, and c represents the speed of light in a vacuum. The speed of light is a constant value, approximately meters per second.
  • This formula shows that energy is directly proportional to mass.
  • It has led to significant advancements in scientific fields, including nuclear physics and cosmology.
Implications on Modern Science:
鈥� Confirmed that mass and energy are interchangeable.
鈥� Deepened our understanding of space and time.
鈥� Paved the way for nuclear technology and cosmology advancements.Understanding these connections helps appreciate the equation's importance in broader scientific theories and real-world applications.

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