Solve for v Ec=1/2mv^2

$E c = \frac{1}{2} m v^{2}$

Step 1

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

$\frac{1}{2} \cdot (m v^{2}) = E c$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (m v^{2})) = 2 (E c)$

Step 3

Simplify both sides of the equation.

Simplify the left side.

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

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

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

$2 (\frac{m}{2} v^{2}) = 2 (E c)$

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

$2 \frac{m v^{2}}{2} = 2 (E c)$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{m v^{2}}{2} = 2 (E c)$

Rewrite the expression.

$m v^{2} = 2 (E c)$

Simplify the right side.

Remove parentheses.

$m v^{2} = 2 E c$

Step 4

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

Divide each term in $m v^{2} = 2 E c$ by $m$ .

$\frac{m v^{2}}{m} = \frac{2 E c}{m}$

Simplify the left side.

Cancel the common factor of $m$ .

Cancel the common factor.

$\frac{m v^{2}}{m} = \frac{2 E c}{m}$

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

$v^{2} = \frac{2 E c}{m}$

Step 5

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

$v = \pm \sqrt{\frac{2 E c}{m}}$

Step 6

Simplify $\pm \sqrt{\frac{2 E c}{m}}$ .

Rewrite $\sqrt{\frac{2 E c}{m}}$ as $\frac{\sqrt{2 E c}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 E c}}{\sqrt{m}}$

Multiply $\frac{\sqrt{2 E c}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 E c}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2 E c}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

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

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

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

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

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

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{{\sqrt{m}}^{2}}$

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

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

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

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

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

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

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{m^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{m^{\frac{2}{2}}}$

Rewrite the expression.

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{m^{1}}$

Simplify.

$v = \pm \frac{\sqrt{2 E c} \sqrt{m}}{m}$

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{2 E c m}}{m}$

Step 7

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.

$v = \frac{\sqrt{2 E c m}}{m}$

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

$v = - \frac{\sqrt{2 E c m}}{m}$

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

$v = \frac{\sqrt{2 E c m}}{m}$

$v = - \frac{\sqrt{2 E c m}}{m}$

$v = \frac{\sqrt{2 E c m}}{m}$

$v = - \frac{\sqrt{2 E c m}}{m}$

Solve for v Ec=1/2mv^2

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