Given the position function $\mathbf{r}(t) = t^2 \mathbf{i} + 4t \mathbf{j} + 6 \ln(t) \mathbf{k}$, let's find the velocity, acceleration, and speed of the particle.

1. Velocity ($\mathbf{v}(t)$)

The velocity vector is the first derivative of the position vector with respect to time $t$: $\mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t)$ Differentiating each component of $\mathbf{r}(t)$: $\mathbf{v}(t) = \frac{d}{dt}(t^2) \mathbf{i} + \frac{d}{dt}(4t) \mathbf{j} + \frac{d}{dt}(6 \ln(t)) \mathbf{k}$ $\mathbf{v}(t) = 2t \mathbf{i} + 4 \mathbf{j} + \frac{6}{t} \mathbf{k}$ Thus, the velocity vector is: $\mathbf{v}(t) = 2t \mathbf{i} + 4 \mathbf{j} + \frac{6}{t} \mathbf{k}$

2. Acceleration ($\mathbf{a}(t)$)

The acceleration vector is the second derivative of the position vector with respect to time $t$, or the first derivative of the velocity vector: $\mathbf{a}(t) = \frac{d}{dt} \mathbf{v}(t)$ Differentiating each component of $\mathbf{v}(t)$: $\mathbf{a}(t) = \frac{d}{dt}(2t) \mathbf{i} + \frac{d}{dt}(4) \mathbf{j} + \frac{d}{dt}\left(\frac{6}{t}\right) \mathbf{k}$ $\mathbf{a}(t) = 2 \mathbf{i} + 0 \mathbf{j} - \frac{6}{t^2} \mathbf{k}$ Thus, the acceleration vector is: $\mathbf{a}(t) = 2 \mathbf{i} - \frac{6}{t^2} \mathbf{k}$

3. Speed ($|\mathbf{v}(t)|$)

The speed is the magnitude of the velocity vector: $|\mathbf{v}(t)| = \sqrt{(2t)^2 + (4)^2 + \left(\frac{6}{t}\right)^2}$ $|\mathbf{v}(t)| = \sqrt{4t^2 + 16 + \frac{36}{t^2}}$ Thus, the speed is: $|\mathbf{v}(t)| = \sqrt{4t^2 + 16 + \frac{36}{t^2}}$

Summary:

• Velocity: $\mathbf{v}(t) = 2t \mathbf{i} + 4 \mathbf{j} + \frac{6}{t} \mathbf{k}$
• Acceleration: $\mathbf{a}(t) = 2 \mathbf{i} - \frac{6}{t^2} \mathbf{k}$
• Speed: $|\mathbf{v}(t)| = \sqrt{4t^2 + 16 + \frac{36}{t^2}}$

Do you have any questions or would you like further clarification?

Here are 5 related questions you might find helpful:

1. How do you find the unit vector of the velocity?
2. What is the significance of speed versus velocity?
3. How would you compute the distance traveled by the particle?
4. What are the initial conditions for velocity and acceleration when $t = 1$?
5. How does the motion change as $t \to \infty$?

Tip: To analyze a particle's motion, always start by finding the velocity and acceleration as functions of time to understand its dynamic behavior.