Trigonometry Examples

$a (k) = \frac{1}{2} \cdot (m v^{2})$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (m v^{2}) = a k$ .

$\frac{1}{2} \cdot (m v^{2}) = a k$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (m v^{2})) = 2 (a k)$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $2 (\frac{1}{2} \cdot (m v^{2}))$ .

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Step 3.1.1.1

Multiply $\frac{1}{2} (m v^{2})$ .

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Step 3.1.1.1.1

Combine $m$ and $\frac{1}{2}$ .

$2 (\frac{m}{2} v^{2}) = 2 (a k)$

Step 3.1.1.1.2

Combine $\frac{m}{2}$ and $v^{2}$ .

$2 \frac{m v^{2}}{2} = 2 (a k)$

Step 3.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.2.1

Cancel the common factor.

$2 \frac{m v^{2}}{2} = 2 (a k)$

Step 3.1.1.2.2

Rewrite the expression.

$m v^{2} = 2 (a k)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Remove parentheses.

$m v^{2} = 2 a k$

Step 4

Divide each term in $m v^{2} = 2 a k$ by $m$ and simplify.

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Step 4.1

Divide each term in $m v^{2} = 2 a k$ by $m$ .

$\frac{m v^{2}}{m} = \frac{2 a k}{m}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $m$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{m v^{2}}{m} = \frac{2 a k}{m}$

Step 4.2.1.2

Divide $v^{2}$ by $1$ .

$v^{2} = \frac{2 a k}{m}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$v = \pm \sqrt{\frac{2 a k}{m}}$

Step 6

Simplify $\pm \sqrt{\frac{2 a k}{m}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{2 a k}{m}}$ as $\frac{\sqrt{2 a k}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 a k}}{\sqrt{m}}$

Step 6.2

Multiply $\frac{\sqrt{2 a k}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 a k}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{\sqrt{2 a k}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

Step 6.3.2

Raise $\sqrt{m}$ to the power of $1$ .

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

Step 6.3.3

Raise $\sqrt{m}$ to the power of $1$ .

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

Step 6.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Step 6.3.5

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{{\sqrt{m}}^{2}}$

Step 6.3.6

Rewrite ${\sqrt{m}}^{2}$ as $m$ .

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Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m}$ as $m^{\frac{1}{2}}$ .

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

Step 6.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

Step 6.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{m^{\frac{2}{2}}}$

Step 6.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{m^{\frac{2}{2}}}$

Step 6.3.6.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{m^{1}}$

Step 6.3.6.5

Simplify.

$v = \pm \frac{\sqrt{2 a k} \sqrt{m}}{m}$

Step 6.4

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{2 a k m}}{m}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$v = \frac{\sqrt{2 a k m}}{m}$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$v = - \frac{\sqrt{2 a k m}}{m}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$v = \frac{\sqrt{2 a k m}}{m}$

$v = - \frac{\sqrt{2 a k m}}{m}$

$v = \frac{\sqrt{2 a k m}}{m}$

$v = - \frac{\sqrt{2 a k m}}{m}$