Questions: While working on a kinematics problem, you derive an expression that has the following units: sqrt(m / ((m / s) / (m / s))^2) Is this expression unitless?

Transcript text: STEM Fluancy Unit 8 Stem Fluency osu.student.stemfluency.com/takequiz?quiz_id=2216\&course_id=71\§ion_id=18 Assignment - Unit 8 Trigonometry - speclal Trig Components Method Work done by a constant angles in degrees $100 \%$ $100 \%$ $100 \%$ Question: While working on a kinematics problem, you derive an expression that has the following units: \[ \sqrt{\frac{\mathrm{m}}{\left(\frac{\mathrm{~m} / \mathrm{s}}{\mathrm{~m} / \mathrm{s}}\right)^{2}}} \] Is this expression unitless?

Solution

Solution Steps

Step 1: Analyze the Expression

The given expression is:

\[ \sqrt{\frac{\mathrm{m}}{\left(\frac{\mathrm{~m} / \mathrm{s}}{\mathrm{~m} / \mathrm{s}}\right)^{2}}} \]

First, let's simplify the denominator inside the square root.

Step 2: Simplify the Denominator

The denominator is:

\[ \left(\frac{\mathrm{~m} / \mathrm{s}}{\mathrm{~m} / \mathrm{s}}\right)^{2} \]

The fraction $\frac{\mathrm{~m} / \mathrm{s}}{\mathrm{~m} / \mathrm{s}}$ simplifies to 1, since the units cancel out:

\[ \frac{\mathrm{~m} / \mathrm{s}}{\mathrm{~m} / \mathrm{s}} = 1 \]

Thus, the denominator becomes:

\[ 1^2 = 1 \]

Step 3: Simplify the Entire Expression

Substituting back into the original expression, we have:

\[ \sqrt{\frac{\mathrm{m}}{1}} = \sqrt{\mathrm{m}} \]

Step 4: Determine if the Expression is Unitless

The expression $\sqrt{\mathrm{m}}$ has units of $\sqrt{\mathrm{m}}$, which is not unitless. A unitless expression would have no units or be dimensionless.