In introductory kinematics, how do you find the magnitude of the average acceleration of an object over a time interval $\Delta t$, given its initial velocity $v_{\mathrm{i}}$ and final velocity $v_{\mathrm{f}}$ in one dimension?

• A. $a=\frac{\Delta t}{v_{\mathrm{f}}-v_{\mathrm{i}}}$.
• B. $a=v_{\mathrm{f}}-v_{\mathrm{i}}$.
• C. $a=\frac{v_{\mathrm{f}}-v_{\mathrm{i}}}{\Delta t}$ and the magnitude is $\left|a\right|$.
• D. $a=\frac{v_{\mathrm{f}}+v_{\mathrm{i}}}{\Delta t}$.

Answer: (C)

Verified step by step guidance

1. Identify the given quantities: initial velocity \(v_i\), final velocity \(v_f\), and the time interval \(\Delta t\) over which the change occurs.
2. Recall the definition of average acceleration \(a\) as the change in velocity divided by the time interval: \[a = \frac{v_f - v_i}{\Delta t}\]
3. Calculate the difference in velocity by subtracting the initial velocity from the final velocity: \[\Delta v = v_f - v_i\]
4. Divide the velocity change \(\Delta v\) by the time interval \(\Delta t\) to find the average acceleration: \[a = \frac{\Delta v}{\Delta t}\]
5. To find the magnitude of the average acceleration, take the absolute value of \(a\): \[|a| = \left| \frac{v_f - v_i}{\Delta t} \right|\]