Question | AP Tutor

7. A particle moves along the \( x \)-axis so that at any time \( t \geq 0 \), its position is given by:

\[ x(t) = -\frac{1}{2} \cos t - 3t \]

What is the acceleration of the particle when \( t = \frac{\pi}{3} \)?

\( -\frac{\sqrt{3}}{4} \)

\( -\frac{1}{4} \)

\( \frac{1}{4} \)

\( \frac{\sqrt{3}}{4} \)

Answer is: option3

\( \frac{1}{4} \)

Solution:

Given Position Function:

\[ x(t) = -\frac{1}{2} \cos t - 3t \]

Step 1: Differentiate to Find Velocity

Velocity function:

\[ v(t) = \frac{1}{2} \sin t - 3 \]

Step 2: Differentiate to Find Acceleration

Acceleration function:

\[ a(t) = \frac{1}{2} \cos t \]

Step 3: Evaluate at \( t = \frac{\pi}{3} \)

\[ a\left(\frac{\pi}{3}\right) = \frac{1}{2} \cos \left(\frac{\pi}{3}\right) \]

\[ = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

Final Answer: \( \frac{1}{4} \)

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