Problem 50 - Elementary Algebra

Question

Use numerical evaluation on the equations. \(E=m c^{2} . \quad\) Find \(E\) if \(m=5\) and \(c=186,000 .\)

Step-by-Step Solution

Answer

Question: Given the equation \(E=mc^2\) where \(E\) is the energy, \(m\) is the mass, and \(c\) is the speed of light, find the value of \(E\) when \(m = 5\) and \(c = 186,000\). Answer: The energy \(E\) is equal to \(172,980,000,000\) when the mass is \(5\) and the speed of light is \(186,000\).

Step 1: Identify the given values

We are given the values \(m = 5\) and \(c = 186,000\).

Step 2: Substitute the given values into the equation

We will now plug the given values of \(m\) and \(c\) into the equation \(E=mc^2\). Doing so, we have: \(E=(5)(186,000)^2\).

Step 3: Evaluate the equation

We can now evaluate the expression to find the value of \(E\): \(E = 5 \times (186,000)^2 = 5 \times 34,596,000,000 = 172,980,000,000\).

Step 4: Write the final answer

The energy \(E\) is equal to \(172,980,000,000\) when the mass is \(5\) and the speed of light is \(186,000\).

Key Concepts

Numerical Evaluation

Let's dive into numerical evaluation, a crucial process in solving physics equations. Often, we start with an equation that includes symbols representing variables. To find a numerical answer, we substitute these symbols with actual values. In our exercise, we use Einstein’s mass-energy equation:

• The mass (\(m\)) is given as \(5\).
• The speed of light (\(c\)) is \(186,000\) miles per second.

By replacing \(m\) and \(c\) with these values, we transform the equation into a form that can be easily solved using arithmetic. This step-by-step approach simplifies complex concepts, making problem-solving approachable and accurate.

Energy Calculation

Energy calculation is a practical application of numerical evaluation. Starting with the equation \(E = mc^2\), we substitute our values:

• \(E = 5 \times (186,000)^2\)

This equation represents Einstein’s principles of energy relating to mass and the speed of light. We proceed by calculating:

• First, evaluate the square: \((186,000)^2 = 34,596,000,000\).
• Then, multiply by the mass: \(5 \times 34,596,000,000\).

This yields the energy \(E = 172,980,000,000\). By following arithmetic operations carefully, we arrive at a precise measurement of energy, linking abstract concepts to real numbers.

Einstein's Mass-Energy Equation

Einstein’s mass-energy equation is a powerful tool in physics. It describes how mass can be converted into energy:\(E = mc^2\).Here,

• \(E\) is energy
• \(m\) is mass
• \(c\) is the speed of light, a constant.

Einstein's insight shows that even a small amount of mass can be transformed into a large amount of energy due to the enormous value of \(c^2\). This concept has vast implications, from nuclear reactions to understanding the universe's energy. By using this equation, we connect the fundamental properties of matter to the immense energy potential it holds. Understanding \(E=mc^2\) enables us to appreciate the deep connections between mass and energy that underpin many scientific phenomena.