Given the vector function $\mathbf{r}(t) = (5t^2 + 5t) \hat{i} + (5t^2 + 5t) \hat{j}$, we are tasked with finding $\mathbf{r}'(t) \cdot \mathbf{r}''(t)$, which is the dot product of the first and second derivatives of the vector function.

Step 1: Differentiate $\mathbf{r}(t)$ to find $\mathbf{r}'(t)$

The vector function is given as: $\mathbf{r}(t) = (5t^2 + 5t) \hat{i} + (5t^2 + 5t) \hat{j}$

Differentiate $\mathbf{r}(t)$ with respect to $t$: $\mathbf{r}'(t) = \frac{d}{dt} \left( (5t^2 + 5t) \hat{i} + (5t^2 + 5t) \hat{j} \right)$

For the $\hat{i}$-component: $\frac{d}{dt}(5t^2 + 5t) = 10t + 5$

For the $\hat{j}$-component: $\frac{d}{dt}(5t^2 + 5t) = 10t + 5$

Thus, we have: $\mathbf{r}'(t) = (10t + 5) \hat{i} + (10t + 5) \hat{j}$

Step 2: Differentiate $\mathbf{r}'(t)$ to find $\mathbf{r}''(t)$

Now, differentiate $\mathbf{r}'(t)$ with respect to $t$: $\mathbf{r}''(t) = \frac{d}{dt} \left( (10t + 5) \hat{i} + (10t + 5) \hat{j} \right)$

For the $\hat{i}$-component: $\frac{d}{dt}(10t + 5) = 10$

For the $\hat{j}$-component: $\frac{d}{dt}(10t + 5) = 10$

Thus, we have: $\mathbf{r}''(t) = 10 \hat{i} + 10 \hat{j}$

Step 3: Compute the dot product $\mathbf{r}'(t) \cdot \mathbf{r}''(t)$

The dot product of two vectors $\mathbf{A} = a_1 \hat{i} + a_2 \hat{j}$ and $\mathbf{B} = b_1 \hat{i} + b_2 \hat{j}$ is given by: $\mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2$

For $\mathbf{r}'(t) = (10t + 5) \hat{i} + (10t + 5) \hat{j}$ and $\mathbf{r}''(t) = 10 \hat{i} + 10 \hat{j}$, we have: $\mathbf{r}'(t) \cdot \mathbf{r}''(t) = (10t + 5)(10) + (10t + 5)(10)$ $\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 100t + 50 + 100t + 50$ $\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 200t + 100$

Final Answer:

$\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 200t + 100$

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How do you compute the cross product between two vector functions?
2. How would the result change if the vector function had a third component $\hat{k}$?
3. What is the physical significance of $\mathbf{r}'(t) \cdot \mathbf{r}''(t)$ in the context of motion?
4. Can you compute $\mathbf{r}(t) \times \mathbf{r}'(t)$ for the same vector function?
5. How do you interpret $\mathbf{r}'(t)$ and $\mathbf{r}''(t)$ geometrically in terms of motion?

Tip: The dot product of two vectors is maximized when the vectors are parallel and minimized when they are perpendicular.