To solve for $v$ in the formula for kinetic energy and complete the equation:

The equation we are solving is:
$\sqrt{K} \cdot 1.4142 \, m^{-\frac{1}{2}} \left( \, \_ \, \right) = \sqrt{m} \cdot v \, m^{-\frac{1}{2}} \left( \, \_ \, \right)$

Step-by-Step Solution

1. The goal is to isolate $v$ on one side of the equation.
2. Notice that both sides involve terms of mass $m$ with exponents and a square root relationship.

The correct multiplier that allows us to cancel terms and solve for $v$ is $m^{\frac{1}{2}}$. Here's why:

• Since $m^{-\frac{1}{2}}$ exists, multiplying by $m^{\frac{1}{2}}$ neutralizes the exponent because:
  $m^{\frac{1}{2}} \cdot m^{-\frac{1}{2}} = m^0 = 1.$

Updated Equation

When you multiply both sides by $m^{\frac{1}{2}}$, the equation simplifies to:
$\sqrt{K} \cdot 1.4142 = v$

Thus, the solution for $v$ is:
$v = \sqrt{K} \cdot 1.4142.$

Follow-Up Questions:

1. What is the role of square roots in solving kinetic energy problems?
2. How can you isolate a variable in equations involving exponents?
3. What does $1.4142$ represent in terms of square roots?
4. Why do we multiply by $m^{\frac{1}{2}}$ to simplify terms involving mass?
5. How would the formula change if solving for $m$ instead of $v$?

Tip:

When solving for variables, identify terms that can cancel out using exponent properties (e.g., $m^{a} \cdot m^{-a} = 1$). This simplifies the equation efficiently.