To calculate the average velocity of the particle over the given time intervals, we use the formula for average velocity:

$v_{\text{avg}} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

where:

• $s(t)$ is the displacement at time $t$.
• $t_1$ and $t_2$ are the initial and final times of the interval.

Displacement formula:

$s(t) = 5 \sin(\pi t) + 2 \cos(\pi t)$

(a) Find the average velocity for the interval [1, 1.01].

1. Calculate $s(1)$: $s(1) = 5 \sin(\pi \cdot 1) + 2 \cos(\pi \cdot 1) = 5 \sin(\pi) + 2 \cos(\pi)$ Since $\sin(\pi) = 0$ and $\cos(\pi) = -1$, we have: $s(1) = 5(0) + 2(-1) = -2$

2. Calculate $s(1.01)$: $s(1.01) = 5 \sin(\pi \cdot 1.01) + 2 \cos(\pi \cdot 1.01)$ We can compute this using approximate values of the sine and cosine functions.

Let's compute the values for both steps.For the interval $[1, 1.01]$:

• $s(1) = -2.00 \, \text{cm}$
• $s(1.01) = -2.16 \, \text{cm}$

The average velocity is approximately:

$v_{\text{avg}} = -15.61 \, \text{cm/s}$

(b) Now, let's compute the average velocity for the interval $[1, 1.001]$:

1. We already have $s(1) = -2$.

2. Now, calculate $s(1.001)$:

$s(1.001) = 5 \sin(\pi \cdot 1.001) + 2 \cos(\pi \cdot 1.001)$

I'll compute this next.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can manually calculate the displacement $s(1.001)$ using the formula:

$s(t) = 5 \sin(\pi t) + 2 \cos(\pi t)$

Once you have that, you can use the formula for average velocity:

$v_{\text{avg}} = \frac{s(1.001) - s(1)}{1.001 - 1}$

Let me know if you'd like any further assistance or explanations! Would you like more help with these calculations?

Here are some related questions:

1. What is the instantaneous velocity at $t = 1$?
2. How do we calculate the maximum speed of the particle?
3. What happens to the average velocity as the time interval decreases?
4. Can we find the time when the particle is at rest?
5. How does changing the coefficients in the equation affect the motion?

Tip: The average velocity formula is similar to calculating the slope between two points on a graph!