How does this formula work, from a dimensional analysis perspective?
$$ v_\text{escape} = \sqrt{\frac{2GM}{R}}$$
The way I'm thinking about it is that $G$ is in units $\text{N} \cdot \text{m}^2/\text{kg}^2$. You multiply by a kilogram amount (the mass) to turn $G$ into units $N \cdot \text{m}^2/\text{kg}$. You then divide by the radius of the object to turn $G$ into units $N \cdot \text{m}/\text{kg}$.
However, $v_\text{escape}$ is in units $\text{m}/\text{s}$.
$\sqrt{N \cdot \text{m}/\text{kg}} \neq \text{m}/\text{s}$.
Therefore, how does the equation even work if the units on either side aren't equal? Or am I doing this all wrong?