**Solve the equation for the indicated variable. Assume no denominator is 0.**

Ve = $$ \frac12 $$mv$$ ^2 $$, for v

The given equation is:

\[
V_e = \frac{1}{2} m v^2
\]

We need to solve for $v$.

### Step-by-step solution:

1. **Isolate $v^2$**:  
   First, multiply both sides of the equation by 2 to get rid of the fraction:
   \[
   2 V_e = m v^2
   \]

2. **Solve for $v^2$**:  
   Divide both sides by $m$:
   \[
   v^2 = \frac{2 V_e}{m}
   \]

3. **Take the square root**:  
   Finally, take the square root of both sides to solve for $v$:
   \[
   v = \sqrt{\frac{2 V_e}{m}}
   \]

Thus, the solution for $v$ is:

\[
v = \sqrt{\frac{2 V_e}{m}}
\]

### Do you have any questions or would you like further details?

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Here are 5 related questions to consider:
1. What would happen if $V_e = 0$ in this equation?
2. How can this equation be applied to calculate velocity in real-world scenarios?
3. How does this equation relate to kinetic energy?
4. Can you solve for $m$ in terms of $V_e$ and $v$?
5. What physical quantity does $V_e$ represent?

**Tip:** When solving for variables under square roots, always consider both the positive and negative roots unless context (like velocity) restricts the solution to positive values.

Solve the equation for the indicated variable. Assume no denominator is 0. Ve = (1/2)mv^2, for v

This detailed guide explains how to solve the equation Ve = (1/2)mv^2 for the variable v, covering algebraic techniques like isolating variables and using square roots. Learn how the equation relates to kinetic energy and understand how to manipulate it to find velocity. Suitable for students in Grades 9-12.

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