To isolate $v$ in the formula for kinetic energy:

$E = \frac{1}{2} mv^2$

Here are the steps:

1. Multiply both sides by 2 to get rid of the fraction: $2E = mv^2$

2. Divide both sides by $m$ to isolate $v^2$: $v^2 = \frac{2E}{m}$

3. Take the square root of both sides to solve for $v$: $v = \sqrt{\frac{2E}{m}}$

Thus, the final equation is: $v = \sqrt{\frac{2E}{m}}$

Would you like any further details or explanations?

Here are 5 related questions to explore:

1. How does the formula change if you include a potential energy term?
2. What does the variable $m$ represent, and how does it affect $v$?
3. What units would $E$, $m$, and $v$ be in for this equation?
4. How can this formula be applied in the context of classical mechanics?
5. What would the formula look like if you wanted to solve for $E$ instead of $v$?

Tip: When dealing with square roots in physics, always consider whether you need both the positive and negative root, especially when dealing with velocities.