https://file.aiquickdraw.com/mathbot/file/1727060239806ykhkfch9.jpg

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 \( -12.1 \, \text{cm/sec} \).

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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.

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 14 cos(t), where x is measured in centimeters and t is measured in seconds. What is the spring's velocity when t = 0 and when t = π/3?

Learn how to calculate the spring's velocity in a harmonic motion where the displacement is x = 14 cos(t). This problem covers key calculus concepts such as differentiating trigonometric functions to find velocity. Suitable for Grades 11-12.

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