Problem 73 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for the indicated variable. Assume no denominators are \(0 .\) $$F=\frac{k M v^{2}}{r}, \quad \text { for } v$$

Step-by-Step Solution

Answer

v = \sqrt{\frac{F \, r}{k M}}

Step 1: Identify the given equation

The equation provided is \[ F = \frac{k M v^{2}}{r} \] and we need to solve for the variable \(v\).

Step 2: Isolate the term containing the variable

Multiply both sides of the equation by \(r\) to isolate the term with \(v^{2}\). This results in \[ F \, r = k M v^{2} \].

Step 3: Solve for the variable squared

Divide both sides by \(k M\) to solve for \(v^{2}\): \[ v^{2} = \frac{F \, r}{k M} \].

Step 4: Take the square root

Take the square root of both sides to solve for \(v\): \[ v = \sqrt{\frac{F \, r}{k M}} \].

Key Concepts

Variable Isolation

In algebra, one of the first steps in solving an equation for a variable is to isolate the term containing that variable. This means rearranging the equation so that the variable you're solving for is on one side of the equation and everything else is on the other side. Consider our exercise example, where we start with the equation \[ F = \frac{kM v^{2}}{r} \]. To isolate the term containing \( v \), we multiply both sides of the equation by \( r \), giving us \[ F \, r = kM v^{2} \]. Now, the term \( v^{2} \) is isolated, meaning the variable \( v \) is now more straightforward to solve for.

Multiplication

Multiplication is often used in the process of isolating a variable. It allows us to clear fractions or to balance the equation. In our example, we used multiplication to eliminate the fraction. Specifically, we multiplied both sides of the equation \[ F = \frac{kM v^{2}}{r} \] by \( r \). This step transforms the equation to \[ F \, r = kM v^{2} \]. Multiplying by \( r \) balances the equation on both sides and helps isolate the variable term, making the next steps easier.

Division

After isolating the term with the variable, our next step is often division. Division helps to further isolate the variable itself. In our equation, after isolating \( kM v^{2} \), we divide both sides by \( kM \): this means \[ v^{2} = \frac{F \, r}{kM} \]. By dividing, we removed the multipliers of the variable term. Now, \( v^{2} \) is isolated, and we can go to the final step to solve for \( v \).

Square Root

The final step in solving for \( v \) in our equation involves taking the square root. Once we have an equation where the variable is squared, such as \[ v^{2} = \frac{F \, r}{kM} \], we take the square root of both sides to solve for \( v \). This gives us \[ v = \underline{\phantom{xxx}} \sqrt{\frac{F \, r}{kM}} \]. Taking the square root effectively 'undoes' the squaring, providing us with the value of \( v \) as needed. Remember, when you take the square root, it's important to consider both positive and negative roots.