Solve for v Ma=1/2*(mv^2)+mgh

$M a = \frac{1}{2} \cdot (m v^{2}) + mg h$

Step 1

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

$\frac{1}{2} \cdot (m v^{2}) + mg h = M a$

Step 2

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

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

$\frac{m}{2} v^{2} + mg h = M a$

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

$\frac{m v^{2}}{2} + mg h = M a$

Step 3

Subtract $mg h$ from both sides of the equation.

$\frac{m v^{2}}{2} = M a - mg h$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{m v^{2}}{2} = 2 (M a - mg h)$

Step 5

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{m v^{2}}{2} = 2 (M a - mg h)$

Rewrite the expression.

$m v^{2} = 2 (M a - mg h)$

Simplify the right side.

Simplify $2 (M a - mg h)$ .

Apply the distributive property.

$m v^{2} = 2 (M a) + 2 (- mg h)$

Multiply $- 1$ by $2$ .

$m v^{2} = 2 M a - 2 mg h$

Step 6

Divide each term in $m v^{2} = 2 M a - 2 mg h$ by $m$ and simplify.

Divide each term in $m v^{2} = 2 M a - 2 mg h$ by $m$ .

$\frac{m v^{2}}{m} = \frac{2 M a}{m} + \frac{- 2 mg h}{m}$

Simplify the left side.

Cancel the common factor of $m$ .

Cancel the common factor.

$\frac{m v^{2}}{m} = \frac{2 M a}{m} + \frac{- 2 mg h}{m}$

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

$v^{2} = \frac{2 M a}{m} + \frac{- 2 mg h}{m}$

Simplify the right side.

Move the negative in front of the fraction.

$v^{2} = \frac{2 M a}{m} - \frac{2 mg h}{m}$

Step 7

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

$v = \pm \sqrt{\frac{2 M a}{m} - \frac{2 mg h}{m}}$

Step 8

Simplify $\pm \sqrt{\frac{2 M a}{m} - \frac{2 mg h}{m}}$ .

Combine the numerators over the common denominator.

$v = \pm \sqrt{\frac{2 M a - 2 mg h}{m}}$

Factor $2$ out of $2 M a - 2 mg h$ .

Factor $2$ out of $2 M a$ .

$v = \pm \sqrt{\frac{2 (M a) - 2 mg h}{m}}$

Factor $2$ out of $- 2 mg h$ .

$v = \pm \sqrt{\frac{2 (M a) + 2 (- mg h)}{m}}$

Factor $2$ out of $2 (M a) + 2 (- mg h)$ .

$v = \pm \sqrt{\frac{2 (M a - mg h)}{m}}$

Rewrite $\sqrt{\frac{2 (M a - mg h)}{m}}$ as $\frac{\sqrt{2 (M a - mg h)}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 (M a - mg h)}}{\sqrt{m}}$

Multiply $\frac{\sqrt{2 (M a - mg h)}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 (M a - mg h)}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2 (M a - mg h)}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

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

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

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

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

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

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{{\sqrt{m}}^{2}}$

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

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

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

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

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

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

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{m^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{m^{\frac{2}{2}}}$

Rewrite the expression.

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{m^{1}}$

Simplify.

$v = \pm \frac{\sqrt{2 (M a - mg h)} \sqrt{m}}{m}$

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{2 (M a - mg h) m}}{m}$

Reorder factors in $\pm \frac{\sqrt{2 (M a - mg h) m}}{m}$ .

$v = \pm \frac{\sqrt{2 m (M a - mg h)}}{m}$

Step 9

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

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

$v = \frac{\sqrt{2 m (M a - mg h)}}{m}$

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

$v = - \frac{\sqrt{2 m (M a - mg h)}}{m}$

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

$v = \frac{\sqrt{2 m (M a - mg h)}}{m}, - \frac{\sqrt{2 m (M a - mg h)}}{m}$

Step 10

Reorder factors in $v = \frac{\sqrt{2 m (M a - mg h)}}{m}, - \frac{\sqrt{2 m (M a - mg h)}}{m}$ .

$v = \frac{\sqrt{2 m (M a - h mg)}}{m}, - \frac{\sqrt{2 m (M a - h mg)}}{m}$

Solve for v Ma=1/2*(mv^2)+mgh

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