Solve for v P=2/3(N/V)1/2mv^2

$P = \frac{2}{3} (\frac{N}{V}) \frac{1}{2} m v^{2}$

Step 1

Rewrite the equation as $\frac{2}{3} \cdot (\frac{N}{V}) \cdot \frac{1}{2} \cdot (m v^{2}) = P$ .

$\frac{2}{3} \cdot (\frac{N}{V}) \cdot \frac{1}{2} \cdot (m v^{2}) = P$

Step 2

Simplify.

Multiply $\frac{2}{3}$ by $\frac{N}{V}$ .

$\frac{2 N}{3 V} \cdot \frac{1}{2} \cdot (m v^{2}) = P$

Multiply $\frac{2 N}{3 V}$ by $\frac{1}{2}$ .

$\frac{2 N}{3 V \cdot 2} \cdot (m v^{2}) = P$

Multiply $2$ by $3$ .

$\frac{2 N}{6 V} \cdot (m v^{2}) = P$

Combine $m$ and $\frac{2 N}{6 V}$ .

$\frac{m (2 N)}{6 V} \cdot v^{2} = P$

Combine $\frac{m (2 N)}{6 V}$ and $v^{2}$ .

$\frac{m (2 N) v^{2}}{6 V} = P$

Remove parentheses.

$\frac{m \cdot 2 N v^{2}}{6 V} = P$

Step 3

Multiply each term in $\frac{m \cdot (2 N v^{2})}{6 V} = P$ by $\frac{6 V}{2 N m}$ to eliminate the fractions.

Multiply each term in $\frac{m \cdot (2 N v^{2})}{6 V} = P$ by $\frac{6 V}{2 N m}$ .

$\frac{m \cdot (2 N v^{2})}{6 V} \cdot \frac{6 V}{2 N m} = P \frac{6 V}{2 N m}$

Simplify the left side.

Combine.

$\frac{m \cdot (2 N v^{2}) (6 V)}{6 V (2 N m)} = P \frac{6 V}{2 N m}$

Cancel the common factor of $m$ .

Cancel the common factor.

$\frac{m \cdot (2 N v^{2}) (6 V)}{6 V (2 N m)} = P \frac{6 V}{2 N m}$

Rewrite the expression.

$\frac{(2 N v^{2}) (6 V)}{6 V (2 N)} = P \frac{6 V}{2 N m}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $(2 N v^{2}) (6 V)$ .

$\frac{2 ((N v^{2}) (6 V))}{6 V (2 N)} = P \frac{6 V}{2 N m}$

Cancel the common factors.

Factor $2$ out of $6 V (2 N)$ .

$\frac{2 ((N v^{2}) (6 V))}{2 (3 V (2 N))} = P \frac{6 V}{2 N m}$

Cancel the common factor.

$\frac{2 ((N v^{2}) (6 V))}{2 (3 V (2 N))} = P \frac{6 V}{2 N m}$

Rewrite the expression.

$\frac{(N v^{2}) (6 V)}{3 V (2 N)} = P \frac{6 V}{2 N m}$

Cancel the common factor of $N$ .

Cancel the common factor.

$\frac{N v^{2} (6 V)}{3 V (2 N)} = P \frac{6 V}{2 N m}$

Rewrite the expression.

$\frac{(v^{2}) (6 V)}{3 V \cdot (2)} = P \frac{6 V}{2 N m}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $(v^{2}) (6 V)$ .

$\frac{3 (v^{2} (2 V))}{3 V \cdot (2)} = P \frac{6 V}{2 N m}$

Cancel the common factors.

Factor $3$ out of $3 V \cdot (2)$ .

$\frac{3 (v^{2} (2 V))}{3 (V \cdot (2))} = P \frac{6 V}{2 N m}$

Cancel the common factor.

$\frac{3 (v^{2} (2 V))}{3 (V \cdot (2))} = P \frac{6 V}{2 N m}$

Rewrite the expression.

$\frac{v^{2} (2 V)}{V \cdot (2)} = P \frac{6 V}{2 N m}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{v^{2} (2 V)}{V \cdot 2} = P \frac{6 V}{2 N m}$

Rewrite the expression.

$\frac{v^{2} (V)}{V} = P \frac{6 V}{2 N m}$

Cancel the common factor of $V$ .

Cancel the common factor.

$\frac{v^{2} V}{V} = P \frac{6 V}{2 N m}$

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

$v^{2} = P \frac{6 V}{2 N m}$

Simplify the right side.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6 V$ .

$v^{2} = P \frac{2 (3 V)}{2 N m}$

Cancel the common factors.

Factor $2$ out of $2 N m$ .

$v^{2} = P \frac{2 (3 V)}{2 (N m)}$

Cancel the common factor.

$v^{2} = P \frac{2 (3 V)}{2 (N m)}$

Rewrite the expression.

$v^{2} = P \frac{3 V}{N m}$

Combine $P$ and $\frac{3 V}{N m}$ .

$v^{2} = \frac{P (3 V)}{N m}$

Move $3$ to the left of $P$ .

$v^{2} = \frac{3 P V}{N m}$

Step 4

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

$v = \pm \sqrt{\frac{3 P V}{N m}}$

Step 5

Simplify $\pm \sqrt{\frac{3 P V}{N m}}$ .

Rewrite $\sqrt{\frac{3 P V}{N m}}$ as $\frac{\sqrt{3 P V}}{\sqrt{N m}}$ .

$v = \pm \frac{\sqrt{3 P V}}{\sqrt{N m}}$

Multiply $\frac{\sqrt{3 P V}}{\sqrt{N m}}$ by $\frac{\sqrt{N m}}{\sqrt{N m}}$ .

$v = \pm \frac{\sqrt{3 P V}}{\sqrt{N m}} \cdot \frac{\sqrt{N m}}{\sqrt{N m}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 P V}}{\sqrt{N m}}$ by $\frac{\sqrt{N m}}{\sqrt{N m}}$ .

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{\sqrt{N m} \sqrt{N m}}$

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

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{\sqrt{N m}}^{1} \sqrt{N m}}$

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

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{\sqrt{N m}}^{1} {\sqrt{N m}}^{1}}$

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

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{\sqrt{N m}}^{1 + 1}}$

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{\sqrt{N m}}^{2}}$

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

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

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{({(N m)}^{\frac{1}{2}})}^{2}}$

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

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{(N m)}^{\frac{1}{2} \cdot 2}}$

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

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{(N m)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{(N m)}^{\frac{2}{2}}}$

Rewrite the expression.

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{{(N m)}^{1}}$

Simplify.

$v = \pm \frac{\sqrt{3 P V} \sqrt{N m}}{N m}$

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{3 P V N m}}{N m}$

Step 6

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.

$v = \frac{\sqrt{3 P V N m}}{N m}$

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

$v = - \frac{\sqrt{3 P V N m}}{N m}$

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

$v = \frac{\sqrt{3 P V N m}}{N m}$

$v = - \frac{\sqrt{3 P V N m}}{N m}$

$v = \frac{\sqrt{3 P V N m}}{N m}$

$v = - \frac{\sqrt{3 P V N m}}{N m}$

Solve for v P=2/3(N/V)1/2mv^2

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