Precalculus Examples

$F = \frac{g M v^{2}}{r}$

Step 1

Rewrite the equation as $\frac{g M v^{2}}{r} = F$ .

$\frac{g M v^{2}}{r} = F$

Step 2

Multiply both sides by $r$ .

$\frac{g M v^{2}}{r} r = F r$

Step 3

Simplify the left side.

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Step 3.1

Cancel the common factor of $r$ .

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Step 3.1.1

Cancel the common factor.

$\frac{g M v^{2}}{r} r = F r$

Step 3.1.2

Rewrite the expression.

$g M v^{2} = F r$

Step 4

Solve for $v$ .

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Step 4.1

Divide each term in $g M v^{2} = F r$ by $g M$ and simplify.

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Step 4.1.1

Divide each term in $g M v^{2} = F r$ by $g M$ .

$\frac{g M v^{2}}{g M} = \frac{F r}{g M}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $g$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{g M v^{2}}{g M} = \frac{F r}{g M}$

Step 4.1.2.1.2

Rewrite the expression.

$\frac{M v^{2}}{M} = \frac{F r}{g M}$

Step 4.1.2.2

Cancel the common factor of $M$ .

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Step 4.1.2.2.1

Cancel the common factor.

$\frac{M v^{2}}{M} = \frac{F r}{g M}$

Step 4.1.2.2.2

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

$v^{2} = \frac{F r}{g M}$

Step 4.2

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

$v = \pm \sqrt{\frac{F r}{g M}}$

Step 4.3

Simplify $\pm \sqrt{\frac{F r}{g M}}$ .

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Step 4.3.1

Rewrite $\sqrt{\frac{F r}{g M}}$ as $\frac{\sqrt{F r}}{\sqrt{g M}}$ .

$v = \pm \frac{\sqrt{F r}}{\sqrt{g M}}$

Step 4.3.2

Multiply $\frac{\sqrt{F r}}{\sqrt{g M}}$ by $\frac{\sqrt{g M}}{\sqrt{g M}}$ .

$v = \pm \frac{\sqrt{F r}}{\sqrt{g M}} \cdot \frac{\sqrt{g M}}{\sqrt{g M}}$

Step 4.3.3

Combine and simplify the denominator.

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Step 4.3.3.1

Multiply $\frac{\sqrt{F r}}{\sqrt{g M}}$ by $\frac{\sqrt{g M}}{\sqrt{g M}}$ .

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{\sqrt{g M} \sqrt{g M}}$

Step 4.3.3.2

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

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{\sqrt{g M}}^{1} \sqrt{g M}}$

Step 4.3.3.3

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

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{\sqrt{g M}}^{1} {\sqrt{g M}}^{1}}$

Step 4.3.3.4

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

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{\sqrt{g M}}^{1 + 1}}$

Step 4.3.3.5

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{\sqrt{g M}}^{2}}$

Step 4.3.3.6

Rewrite ${\sqrt{g M}}^{2}$ as $g M$ .

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Step 4.3.3.6.1

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

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{({(g M)}^{\frac{1}{2}})}^{2}}$

Step 4.3.3.6.2

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

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{(g M)}^{\frac{1}{2} \cdot 2}}$

Step 4.3.3.6.3

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

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{(g M)}^{\frac{2}{2}}}$

Step 4.3.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.3.6.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{(g M)}^{\frac{2}{2}}}$

Step 4.3.3.6.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{{(g M)}^{1}}$

Step 4.3.3.6.5

Simplify.

$v = \pm \frac{\sqrt{F r} \sqrt{g M}}{g M}$

Step 4.3.4

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{F r g M}}{g M}$

Step 4.4

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

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Step 4.4.1

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

$v = \frac{\sqrt{F r g M}}{g M}$

Step 4.4.2

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

$v = - \frac{\sqrt{F r g M}}{g M}$

Step 4.4.3

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

$v = \frac{\sqrt{F r g M}}{g M}$

$v = - \frac{\sqrt{F r g M}}{g M}$

$v = \frac{\sqrt{F r g M}}{g M}$

$v = - \frac{\sqrt{F r g M}}{g M}$

$v = \frac{\sqrt{F r g M}}{g M}$

$v = - \frac{\sqrt{F r g M}}{g M}$