Problem 63 - College Algebra

Question

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(E=m c^{2}\) for \(m\)

Step-by-Step Solution

Answer

The solution for \(m\) in the given formula \(E=mc^{2}\) is \(m=\frac{E}{c^{2}}\).

Step 1: Identify the formula and the variable to solve

The given formula is \(E=mc^{2}\), where E represents energy, m is mass, and c is the speed of light (a constant value). The goal is to solve this equation for m.

Step 2: Rewrite the formula to isolate \(m\)

In order to isolate m, we should divide both sides of the equation by \(c^{2}\). By doing so, we will get \(m=\frac{E}{c^{2}}\). This is our solution.

Step 3: Check the solution

To confirm we've solved for m correctly, substitute m back into the original formula, using \(m=\frac{E}{c^{2}}\). The reliability of the equation remains the same, proving that the solution is correct.

Key Concepts

Formula Rearrangement

Rearranging formulas is a fundamental skill in algebra. It involves manipulating an equation to express one variable in terms of others. This process often requires the use of basic arithmetic operations, such as addition, subtraction, multiplication, and division.
In the context of the equation from Einstein's famous formula, rearranging means isolating one variable on one side of the equation. This makes it easier to see the relationship between the variables involved. For the equation \(E = mc^2\), we want to find the mass \(m\). This means we should arrange the formula so \(m\) is by itself on one side.

• Start with the original formula \(E = mc^2\).
• Identify \(m\) as the variable to isolate.
• Divide both sides by \(c^2\) to get \(m = \frac{E}{c^2}\).

Now, we have successfully rearranged the formula to solve for mass, and this rearranged formula can be used whenever you know the energy and need to find the mass.

Solving Equations

Solving equations is the process of finding the value of unknown variables that satisfy a given equation. In this exercise, the goal was to solve for \(m\) in Einstein's mass-energy equivalence formula. Let's break down how solving for the variable works:
To find \(m\), notice that it's multiplied by \(c^2\). To remove \(c^2\) from \(m\), perform the inverse operation. Specifically, in this case, divide both sides of the equation by \(c^2\), the factor multiplying \(m\) in the equation \(E = mc^2\).
This division yields the expression \(m = \frac{E}{c^2}\).

• Inverse operations like division help isolate variables.
• This step is crucial for expressing one variable in terms of others.

By solving equations this way, you can easily find unknown values when other elements are known. Make sure to check your solution by plugging it back into the original formula to confirm its correctness.

Einstein's Mass-Energy Equivalence

Einstein's mass-energy equivalence, expressed in the equation \(E=mc^2\), is a groundbreaking concept in physics. It illustrates the principle that mass and energy are interchangeable, revealing that they are two forms of the same thing.
This formula highlights a vital relationship:

• \(E\) represents energy, usually in joules.
• \(m\) signifies mass, typically in kilograms.
• \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\) meters per second.

The formula implies that even small amounts of mass can be transformed into enormous energy, given that \(c^2\) is a colossal number. This concept has profound implications in various fields, ranging from nuclear energy applications to understanding cosmic phenomena.
Understanding this equivalence is crucial as it helps explain how energy can be released or absorbed during nuclear reactions and why incredibly dense objects like black holes have such powerful gravitational pulls. By comprehending this formula, the massive and miniature alike become clearer, and the fundamental unity of physics is revealed.