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Chapter 2: Problem 45

Solve each formula for the indicated variable. \( K=\frac{w v^{2}}{2 g}\) for \(w\)

Short Answer

Expert verified
\(w = \frac{2gK}{v^2}\)

Step by step solution

01

Understand the given equation

The formula given is \(K=\frac{w v^{2}}{2 g}\). The task is to solve for \(w\) meaning we want to arrange the equation in such a way that \(w\) is alone on one side of the equation.
02

Multiply through by \(2g\)

By multiplying each side of the equation by \(2g\), we can get rid of the fraction. This simplifies the formula greatly, yielding: \(2gK = wv^2\).
03

Solve for \(w\)

Finally, divide both sides by \(v^2\) to isolate \(w\) on one side of the equation. This gives us: \(w = \frac{2gK}{v^2}\)

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Key Concepts

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

Formula Manipulation
Formula manipulation is a technique in algebra that involves rearranging an equation to highlight or solve for a specific variable. For example, given the formula \( K = \frac{wv^2}{2g} \), we might need to solve for \( w \).
This involves altering the formula's structure. By multiplying both sides by \( 2g \), we eliminate the fraction. This gives us \( 2gK = wv^2 \).
Formula manipulation helps make the equations simpler or in a desirable form.
  • Identify which part of the formula needs alteration.
  • Select operations (like multiplication or division) that simplify or reorder the formula.
This process focuses on logical steps and proper property utilization (like inverse operations) to achieve desired changes.
Isolating Variables
Isolating a variable means rearranging an equation so that the variable of interest stands alone on one side. Suppose we start from \( 2gK = wv^2 \), our goal is to leave \( w \) isolated.
To achieve isolation, we can perform operations like division or subtraction.
  • Operations should reverse or cancel out other parts so that the variable is left solo.
  • From \( 2gK = wv^2 \), dividing by \( v^2 \) isolates \( w \).
The result, \( w = \frac{2gK}{v^2} \), demonstrates a critical aspect of solving algebraic equations: being able to isolate variables effectively facilitates further problem solving.
Algebraic Equations
Algebraic equations are statements of equality featuring variables and constants. They serve as mathematical models for real-world scenarios, often requiring us to find unknowns (variables like \( w \)).
These equations allow us to manipulate variables and constants using arithmetic operations.
  • Balance is key: whatever you do to one side, do to the other.
  • Common operations include addition, subtraction, multiplication, and division.
The equation \( K = \frac{wv^2}{2g} \) illustrates the power of algebraic equations to model relations. Solving it for \( w \) showcases typical steps of equation manipulation and the algebraic understanding necessary to extract values for specific variables.

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