The problem describes the motion of a weight attached to a spring, where the displacement is given by:

$x(t) = 14 \cos(t)$

This is a simple harmonic motion, and the velocity function can be derived by differentiating the displacement $x(t)$ with respect to time $t$.

(a) Finding the spring's velocity when $t = 0$ and $t = \frac{\pi}{3}$

1. Differentiate the displacement function:

The velocity $v(t)$ is the derivative of $x(t)$ with respect to $t$: $v(t) = \frac{dx}{dt} = -14 \sin(t)$

2. (b) Velocity at $t = 0$:

Substitute $t = 0$ into the velocity function: $v(0) = -14 \sin(0) = 0 \, \text{cm/sec}$ This confirms the answer provided in the image, which is $0 \, \text{cm/sec}$.

3. Velocity at $t = \frac{\pi}{3}$:

Substitute $t = \frac{\pi}{3}$ into the velocity function: $v\left(\frac{\pi}{3}\right) = -14 \sin\left(\frac{\pi}{3}\right)$

Since $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$, we get: $v\left(\frac{\pi}{3}\right) = -14 \times \frac{\sqrt{3}}{2} = -7\sqrt{3} \, \text{cm/sec}$ Numerically, this is approximately: $v\left(\frac{\pi}{3}\right) \approx -7 \times 1.732 = -12.1 \, \text{cm/sec}$

Therefore, the velocity at $t = \frac{\pi}{3}$ is approximately $-12.1 \, \text{cm/sec}$.

Would you like more details on any specific part of the solution?

Here are 5 follow-up questions to consider:

1. What is the maximum velocity of the spring's motion?
2. How would the velocity change if the displacement function were $x(t) = 10 \cos(2t)$?
3. What is the period of this spring's motion?
4. How does changing the amplitude affect the velocity at different times?
5. What is the spring's velocity at $t = \pi/2$?

Tip: Remember, velocity is always the derivative of the displacement function when dealing with harmonic motion.