he displacement (in centimeters) of a particle s moving back and forth along a straight line is given by the equation 
s = 5 sin(𝜋t) + 5 cos(𝜋t),
 where t is measured in seconds. (Round your answers to two decimal places.)
(a)
Find the average velocity during each time period.
[1, 1.001]

To find the average velocity of the particle during a given time interval, we use the formula for the average velocity:

\[
\text{Average velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]

where \( s(t) \) is the displacement function, \( t_1 \) is the starting time, and \( t_2 \) is the ending time.

### Given equation of displacement:
\[
s(t) = 5 \sin(\pi t) + 5 \cos(\pi t)
\]

The time interval is [1, 1.001], so \( t_1 = 1 \) and \( t_2 = 1.001 \).

### Step 1: Calculate \( s(1) \)
Substitute \( t = 1 \) into the displacement equation:

\[
s(1) = 5 \sin(\pi \cdot 1) + 5 \cos(\pi \cdot 1)
\]

### Step 2: Calculate \( s(1.001) \)
Substitute \( t = 1.001 \) into the displacement equation:

\[
s(1.001) = 5 \sin(\pi \cdot 1.001) + 5 \cos(\pi \cdot 1.001)
\]

### Step 3: Compute the average velocity
We can now use the values of \( s(1) \) and \( s(1.001) \) to calculate the average velocity.

Let me calculate these values for you.The displacement values are as follows:

- \( s(1) = -5.00 \, \text{cm} \)
- \( s(1.001) = -5.02 \, \text{cm} \)

Thus, the average velocity over the time interval [1, 1.001] is approximately:

\[
\text{Average velocity} = -15.68 \, \text{cm/s}
\]

Would you like more details or have any questions? Here are some related questions to expand on this topic:

1. How is average velocity different from instantaneous velocity?
2. What is the significance of the negative sign in velocity?
3. How do you find the instantaneous velocity at a specific time for this displacement function?
4. How does the particle's velocity change over time in this scenario?
5. What would be the average velocity for a larger interval, such as [1, 2]?

**Tip:** The average velocity gives insight into the overall change in position during a time interval, while instantaneous velocity captures how fast an object is moving at any single moment.

The displacement (in centimeters) of a particle s moving back and forth along a straight line is given by the equation s = 5 sin(𝜋t) + 5 cos(𝜋t), where t is measured in seconds. (Round your answers to two decimal places.) (a) Find the average velocity during each time period. [1, 1.001]

Find the average velocity of a particle moving along a straight line, described by the equation s = 5 sin(πt) + 5 cos(πt). This solution uses the kinematics formula for average velocity over the time interval [1, 1.001], with detailed steps for calculating displacement and velocity. Suitable for advanced high school or undergraduate students.

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