To determine when the speed of the particle is at a minimum, we need to follow these steps:

1. Find the Velocity Vector:

The velocity vector $\mathbf{v}(t)$ is the derivative of the position vector $\mathbf{r}(t)$ with respect to time $t$.

Given: $\mathbf{r}(t) = \langle t^2, 9t, t^2 - 16t \rangle$

The velocity vector $\mathbf{v}(t)$ is: $\mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t) = \langle 2t, 9, 2t - 16 \rangle$

2. Find the Speed:

The speed of the particle is the magnitude of the velocity vector $\mathbf{v}(t)$.

$\text{Speed} = \|\mathbf{v}(t)\| = \sqrt{(2t)^2 + 9^2 + (2t - 16)^2}$

Simplifying this expression:

$\|\mathbf{v}(t)\| = \sqrt{4t^2 + 81 + (2t - 16)^2}$

Expanding and simplifying further:

$(2t - 16)^2 = 4t^2 - 64t + 256$

So, the speed function becomes:

$\|\mathbf{v}(t)\| = \sqrt{4t^2 + 81 + 4t^2 - 64t + 256} = \sqrt{8t^2 - 64t + 337}$

3. Minimize the Speed:

To find when the speed is a minimum, we need to minimize the function $\sqrt{8t^2 - 64t + 337}$.

This is equivalent to minimizing the expression inside the square root:

$f(t) = 8t^2 - 64t + 337$

4. Find the Critical Points:

Take the derivative of $f(t)$ with respect to $t$ and set it equal to zero:

$f'(t) = 16t - 64$ $16t - 64 = 0$

Solve for $t$:

$t = 4$

5. Verify Minimum:

To verify that $t = 4$ gives a minimum, we check the second derivative:

$f''(t) = 16$

Since $f''(t) > 0$, the function $f(t)$ is concave up at $t = 4$, confirming a minimum at this point.

Conclusion:

The speed of the particle is at a minimum when $t = 4$.

---

Would you like further details on any of these steps, or do you have any questions?

Here are some related questions:

1. How do you find the position vector from a velocity vector?
2. What does the minimum speed of a particle indicate about its motion?
3. How would the process change if the velocity function was given in a different form?
4. Can the speed of the particle be negative?
5. What physical situations correspond to finding a minimum speed?

Tip: Minimizing the magnitude of a vector often involves minimizing the square of the magnitude to avoid dealing with the square root directly.