Problem 65 Solve formula for the indicated ... [FREE SOLUTION]

Chapter 8: Problem 65

Solve formula for the indicated variable. \(V=\sqrt{\frac{2 K}{m}}\) for \(K\)

Short Answer

Expert verified

K=\frac{m V^2}{2}

Step by step solution

Square Both Sides

The equation is given as \[ V = \sqrt{\frac{2K}{m}} \] To eliminate the square root, square both sides of the equation. \[ V^2 = \left( \sqrt{\frac{2K}{m}} \right)^2 \] This simplifies to \[ V^2 = \frac{2K}{m} \]

Isolate the Variable K

Now, multiply both sides by \(m\) to isolate the term involving \(K\) on one side of the equation \[ m V^2 = 2K \]

Solve for K

Finally, divide both sides by 2 to solve for \(K\) \[ K = \frac{m V^2}{2} \]

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Key Concepts

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

Isolate the Variable

When solving equations, we often need to focus on one variable. This means getting that variable by itself on one side of the equation.
In our exercise, we start with the equation:

i.e. \( V = \sqrt{\frac{2K}{m}} \)

We want to solve for \( K \). That means we need to change the equation so \( K \) is by itself on one side.
To do this, we follow a systematic approach, making the problem simpler step by step.

Identify the variable you want to isolate. In this case, it's \( K \).
Use algebraic techniques to get rid of other elements around it. Here, we will eventually get rid of the square root and other terms.

Squaring Both Sides

The equation given has a square root. To remove the square root, we can square both sides of the equation.
This is our original equation: \( V = \sqrt{\frac{2K}{m}} \)

When we square both sides, we eliminate the square root on the right side. This is what it looks like:

i.e.
\( V^2 = \left( \sqrt{\frac{2K}{m}} \right)^2 \)
This simplifies to: \( V^2 = \frac{2K}{m} \)

Squaring both sides is a crucial step. It changes a more complex equation into something simpler and easier to work with.
Remember to always perform the same operation on both sides of the equation to keep it balanced.

Algebraic Manipulation

Once you have a simpler form, use algebraic manipulation to solve for the variable.
We have: \( V^2 = \frac{2K}{m} \)
Now, we want to isolate \( K \). To do this:

Multiply both sides by \( m \) to move \( m \) to the left side:

\( m V^2 = 2K \)

The last step is to divide both sides by 2:

\( K = \frac{m V^2}{2} \)

At every step, we use basic algebra. From multiplying both sides to eliminating terms, this is algebraic manipulation.
It's all about making the equation simpler, until the variable you want is isolated and solved.

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91影视

Solve formula for the indicated variable. \(V=\sqrt{\frac{2 K}{m}}\) for \(K\)

Short Answer

Step by step solution

Square Both Sides

Isolate the Variable K

Solve for K

Key Concepts

Isolate the Variable

Squaring Both Sides

Algebraic Manipulation

One App. One Place for Learning.

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