To find the average velocity of the particle between $t = 1.00 \, \text{s}$ and $t = 3.00 \, \text{s}$, we first need to determine the position vector $\vec{r}(t)$ at these two times and then use the formula for average velocity.

Given Position Vector

The position of the particle as a function of time is given by: $\vec{r}(t) = (9.60t \, \hat{i} + 8.85 \, \hat{j} - 1.00t^2 \, \hat{k}) \, \text{m}$

Step 1: Calculate the position vector at $t = 1.00 \, \text{s}$

$\vec{r}(1.00) = (9.60(1.00) \, \hat{i} + 8.85 \, \hat{j} - 1.00(1.00)^2 \, \hat{k}) \, \text{m}$ $\vec{r}(1.00) = (9.60 \, \hat{i} + 8.85 \, \hat{j} - 1.00 \, \hat{k}) \, \text{m}$

Step 2: Calculate the position vector at $t = 3.00 \, \text{s}$

$\vec{r}(3.00) = (9.60(3.00) \, \hat{i} + 8.85 \, \hat{j} - 1.00(3.00)^2 \, \hat{k}) \, \text{m}$ $\vec{r}(3.00) = (28.80 \, \hat{i} + 8.85 \, \hat{j} - 9.00 \, \hat{k}) \, \text{m}$

Step 3: Compute the displacement vector $\Delta \vec{r}$

The displacement vector is given by: $\Delta \vec{r} = \vec{r}(3.00) - \vec{r}(1.00)$ $\Delta \vec{r} = (28.80 \, \hat{i} + 8.85 \, \hat{j} - 9.00 \, \hat{k}) - (9.60 \, \hat{i} + 8.85 \, \hat{j} - 1.00 \, \hat{k})$ $\Delta \vec{r} = (28.80 - 9.60) \, \hat{i} + (8.85 - 8.85) \, \hat{j} + (-9.00 + 1.00) \, \hat{k}$ $\Delta \vec{r} = (19.20 \, \hat{i} + 0 \, \hat{j} - 8.00 \, \hat{k}) \, \text{m}$

Step 4: Calculate the average velocity $\vec{v}_{\text{avg}}$

The average velocity is given by: $\vec{v}_{\text{avg}} = \frac{\Delta \vec{r}}{\Delta t}$ where $\Delta t = 3.00 \, \text{s} - 1.00 \, \text{s} = 2.00 \, \text{s}$.

$\vec{v}_{\text{avg}} = \frac{(19.20 \, \hat{i} + 0 \, \hat{j} - 8.00 \, \hat{k})}{2.00} \, \text{m/s}$ $\vec{v}_{\text{avg}} = (9.60 \, \hat{i} + 0 \, \hat{j} - 4.00 \, \hat{k}) \, \text{m/s}$

Final Answer

The average velocity of the particle between $t = 1.00 \, \text{s}$ and $t = 3.00 \, \text{s}$ is: $\vec{v}_{\text{avg}} = 9.60 \, \hat{i} - 4.00 \, \hat{k} \, \text{m/s}$

Would you like further details or have any questions about this problem? Here are some related questions:

1. How would you determine the instantaneous velocity at $t = 2.00 \, \text{s}$?
2. What would be the particle's displacement vector from $t = 0 \, \text{s}$ to $t = 4.00 \, \text{s}$?
3. How does the position vector change if the coefficient of $t^2$ in the $\hat{k}$ direction were doubled?
4. What is the significance of the $\hat{j}$ component being constant in this problem?
5. How would the average speed differ from the average velocity in this case?

Tip: When dealing with vectors, remember that each component of the vector should be treated independently in calculations like addition, subtraction, and differentiation.