We are given the acceleration vector function:

$a(t) = 5\mathbf{i} + 4\mathbf{j}$

We also have the initial velocity and position:

$v(0) = k$ $r(0) = \mathbf{i}$

Step 1: Find the Velocity Function $v(t)$

Since acceleration is the derivative of velocity, we integrate $a(t)$ to find $v(t)$:

$v(t) = \int a(t) \, dt$

$v(t) = \int (5\mathbf{i} + 4\mathbf{j}) \, dt$

$v(t) = 5t\mathbf{i} + 4t\mathbf{j} + \mathbf{C}$

where $\mathbf{C}$ is the constant of integration.

Using the initial condition $v(0) = k$, we substitute $t = 0$:

$k = 5(0) \mathbf{i} + 4(0) \mathbf{j} + \mathbf{C}$

$\mathbf{C} = k$

Thus, the velocity function is:

$v(t) = (5t + C_1) \mathbf{i} + (4t + C_2) \mathbf{j}$

where $k = C_1 \mathbf{i} + C_2 \mathbf{j}$.

So, in component form:

$v(t) = (5t + C_1) \mathbf{i} + (4t + C_2) \mathbf{j}$

Would you like me to proceed with finding $r(t)$? Let me know if you need any clarifications.

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Related Questions:

1. How do we find the position function $r(t)$ given $v(t)$?
2. What is the physical meaning of the integration constants $C_1$ and $C_2$?
3. How would the solution change if acceleration were a function of $t$?
4. How do we interpret the motion described by this velocity function?
5. What is the significance of the initial position in solving this problem?

Tip: When integrating vector functions, integrate each component separately, then use initial conditions to solve for integration constants.