To determine which formula gives the velocity [tex]\( v \)[/tex] of the car in its simplest form, we will start with the provided equation and simplify it step-by-step.
We are given:
[tex]\[ v = \sqrt{\frac{2k}{m}} \][/tex]
Now, let's work on rationalizing the denominator and simplifying the expression.
### Step-by-Step Simplification:
1. Start with the given equation:
[tex]\[ v = \sqrt{\frac{2k}{m}} \][/tex]
2. Rewrite the square root of a fraction as a fraction of square roots:
[tex]\[ v = \frac{\sqrt{2k}}{\sqrt{m}} \][/tex]
3. To rationalize the denominator, multiply the numerator and the denominator by [tex]\(\sqrt{m}\)[/tex]:
[tex]\[ v = \frac{\sqrt{2k} \cdot \sqrt{m}}{\sqrt{m} \cdot \sqrt{m}} \][/tex]
4. Simplify the denominator:
[tex]\[ \sqrt{m} \cdot \sqrt{m} = m \][/tex]
Therefore, our expression now is:
[tex]\[ v = \frac{\sqrt{2k} \cdot \sqrt{m}}{m} \][/tex]
5. Combine the square roots in the numerator:
[tex]\[ \sqrt{2k} \cdot \sqrt{m} = \sqrt{2km} \][/tex]
Therefore:
[tex]\[ v = \frac{\sqrt{2km}}{m} \][/tex]
We now have the formula in its simplest form:
[tex]\[ v = \frac{\sqrt{2km}}{m} \][/tex]
### Verifying the Options:
Upon comparing this simplified form with the provided options:
1. [tex]\( v = \frac{\sqrt{2km}}{2m} \)[/tex]
2. [tex]\( v = \frac{\sqrt{2km}}{m} \)[/tex]
3. [tex]\( v = \frac{\sqrt{2k}}{m} \)[/tex]
Clearly, the correct option is:
[tex]\[ \boxed{v = \frac{\sqrt{2km}}{m}} \][/tex]
