To find the velocity vector $\mathbf{v}(t)$ and the position vector $\mathbf{c}(t)$, we start with the given acceleration vector $\mathbf{a}(t) = (0, 0, 6)$.

Step 1: Find the velocity vector $\mathbf{v}(t)$

The velocity vector is found by integrating the acceleration vector with respect to time $t$:

$\mathbf{v}(t) = \int \mathbf{a}(t) \, dt = \int (0, 0, 6) \, dt = (0, 0, 6t) + \mathbf{C}$

where $\mathbf{C}$ is the constant of integration. We know the initial velocity:

$\mathbf{v}(0) = (1, 1, -2)$

Substituting $t = 0$:

$(0, 0, 0) + \mathbf{C} = (1, 1, -2) \implies \mathbf{C} = (1, 1, -2)$

Thus, the velocity vector is:

$\mathbf{v}(t) = (1, 1, 6t - 2)$

Step 2: Find the position vector $\mathbf{c}(t)$

Next, we integrate the velocity vector to find the position vector:

$\mathbf{c}(t) = \int \mathbf{v}(t) \, dt = \int (1, 1, 6t - 2) \, dt = (t, t, 3t^2 - 2t) + \mathbf{D}$

where $\mathbf{D}$ is another constant of integration. We know the initial position:

$\mathbf{c}(0) = (3, 4, 0)$

Substituting $t = 0$:

$(0, 0, 0) + \mathbf{D} = (3, 4, 0) \implies \mathbf{D} = (3, 4, 0)$

Thus, the position vector is:

$\mathbf{c}(t) = (t + 3, t + 4, 3t^2 - 2t)$

Step 3: Determine when the particle touches the xy-plane

The particle touches the xy-plane when the $z$-coordinate of the position vector $\mathbf{c}(t)$ is zero:

$3t^2 - 2t = 0$

Factoring out $t$:

$t(3t - 2) = 0$

This gives us two solutions:

1. $t = 0$
2. $3t - 2 = 0 \implies t = \frac{2}{3}$

Summary

The velocity vector is:

$\mathbf{v}(t) = (1, 1, 6t - 2)$

The position vector is:

$\mathbf{c}(t) = (t + 3, t + 4, 3t^2 - 2t)$

The particle touches the xy-plane at:

• $t = 0$ (initial position)
• $t = \frac{2}{3}$

If you have any questions or need further details, feel free to ask! Here are some related questions for further exploration:

1. What is the acceleration vector when $t = \frac{2}{3}$?
2. How would the motion change if the initial velocity were different?
3. What is the path traced by the particle in the xy-plane?
4. How would the position vector change with a different acceleration?
5. Can you find the time when the particle reaches a certain height?

Tip: Always check your integration constants with initial conditions to ensure accuracy.