To find the velocity vector $\mathbf{v}(t)$ and the position vector $\mathbf{c}(t)$ given the acceleration vector $\mathbf{a}(t) = (0, 0, 6)$, we will integrate the acceleration function step by step.

1. Finding the Velocity Vector: The velocity $\mathbf{v}(t)$ is the integral of the acceleration $\mathbf{a}(t)$:

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

   Where $\mathbf{C}$ is the constant of integration. To find $\mathbf{C}$, we use the initial condition for velocity $\mathbf{v}(0) = (1, 1, -2)$:

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

   Therefore, the velocity vector is:

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

2. Finding the Position Vector: The position $\mathbf{c}(t)$ is the integral of the velocity $\mathbf{v}(t)$:

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

   Where $\mathbf{D}$ is another constant of integration. Using the initial position $\mathbf{c}(0) = (3, 4, 0)$:

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

   Thus, the position vector is:

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

3. Finding When the Particle Touches the xy-plane: The particle touches the xy-plane when the z-component of the position vector is zero:

   $3t^2 - 2t = 0$

   Factoring gives:

   $t(3t - 2) = 0$

   Thus, the solutions are:

   $t = 0 \quad \text{or} \quad 3t - 2 = 0 \implies t = \frac{2}{3}$

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 $t = 0$ and $t = \frac{2}{3}$.

Do you want further details on any part of this solution? Here are some related questions you might consider:

1. How do you derive the equations of motion from acceleration?
2. What do the terms in the velocity vector represent physically?
3. How would the results change with a different initial velocity?
4. Can you provide a graphical representation of the motion?
5. What are the implications of the particle's motion after it touches the xy-plane?

Tip: Always check initial conditions when integrating to find constants of integration.