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.