Solve each equation for the specified variable. (Assume no denominators are 0.) F = kMv²/r , for v

Verified step by step guidance

1. Start with the given formula: \(F = \frac{k M v^2}{r}\).
2. Multiply both sides of the equation by \(r\) to eliminate the denominator: \(F r = k M v^2\).
3. Isolate \(v^2\) by dividing both sides by \(k M\): \(\frac{F r}{k M} = v^2\).
4. Take the square root of both sides to solve for \(v\): \(v = \pm \sqrt{\frac{F r}{k M}}\).
5. Since \(v\) represents a variable that might be positive or negative depending on context, consider the appropriate sign based on the problem's physical context.

Key Concepts

Isolating a Variable in an Equation

Isolating a variable means rewriting the equation so that the variable appears alone on one side. This involves using inverse operations like addition, subtraction, multiplication, division, and applying algebraic properties to both sides equally. The goal is to express the specified variable explicitly in terms of the other variables.

Algebraic Manipulation with Fractions

When variables appear in denominators or as part of fractions, careful manipulation is needed to avoid division by zero and to correctly isolate the variable. Multiplying both sides by the denominator or using cross-multiplication helps eliminate fractions and simplify the equation.

Understanding the Given Formula and Variables

Recognizing the roles of each variable in the formula F = kMv^2/r is essential. Here, F, k, M, v, and r represent quantities related by multiplication and division. Knowing how to treat constants and variables helps in rearranging the formula to solve for the specified variable, v.