To solve the problem in the image, we are given the acceleration vector $a(t) = 8\hat{i} + 3\hat{j}$, the initial velocity $v(0) = \hat{k}$, and the initial position $r(0) = \hat{i}$.

Step 1: Finding the velocity vector $v(t)$

The velocity vector is the integral of the acceleration vector. Mathematically,

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

Since $a(t) = 8\hat{i} + 3\hat{j}$, we can integrate each component with respect to time $t$:

$v(t) = \int (8\hat{i} + 3\hat{j}) \, dt$

1. The integral of $8\hat{i}$ with respect to $t$ is $8t\hat{i}$.
2. The integral of $3\hat{j}$ with respect to $t$ is $3t\hat{j}$.

Thus, the velocity vector becomes:

$v(t) = 8t\hat{i} + 3t\hat{j} + C$

where $C$ is the constant of integration, which can be found using the initial condition $v(0) = \hat{k}$. Substituting $t = 0$ into the velocity equation:

$\hat{k} = v(0) = 8(0)\hat{i} + 3(0)\hat{j} + C$

$C = \hat{k}$

Therefore, the velocity vector is:

$v(t) = 8t\hat{i} + 3t\hat{j} + \hat{k}$

Step 2: Finding the position vector $r(t)$

The position vector is the integral of the velocity vector:

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

Substitute $v(t) = 8t\hat{i} + 3t\hat{j} + \hat{k}$:

$r(t) = \int (8t\hat{i} + 3t\hat{j} + \hat{k}) \, dt$

1. The integral of $8t\hat{i}$ with respect to $t$ is $4t^2\hat{i}$.
2. The integral of $3t\hat{j}$ with respect to $t$ is $\frac{3}{2}t^2\hat{j}$.
3. The integral of $\hat{k}$ with respect to $t$ is $t\hat{k}$.

Thus, the position vector becomes:

$r(t) = 4t^2\hat{i} + \frac{3}{2}t^2\hat{j} + t\hat{k} + D$

where $D$ is the constant of integration, which can be found using the initial condition $r(0) = \hat{i}$. Substituting $t = 0$ into the position equation:

$\hat{i} = r(0) = 4(0)^2\hat{i} + \frac{3}{2}(0)^2\hat{j} + 0\hat{k} + D$

$D = \hat{i}$

Therefore, the position vector is:

$r(t) = 4t^2\hat{i} + \frac{3}{2}t^2\hat{j} + t\hat{k} + \hat{i}$

So, the final solution for the velocity and position vectors is:

$v(t) = 8t\hat{i} + 3t\hat{j} + \hat{k}$ $r(t) = (4t^2 + 1)\hat{i} + \frac{3}{2}t^2\hat{j} + t\hat{k}$

Would you like more details on any part of this solution?

5 Related Questions:

1. How do we derive the velocity vector from an acceleration vector in general?
2. What is the physical interpretation of the integration constant $C$ in velocity?
3. How does the initial condition for position help determine the constant $D$?
4. Can this method be applied if acceleration is time-dependent (e.g., $a(t) = t^2\hat{i}$)?
5. What are the implications of a constant acceleration vector in 3D motion?

Tip:

Always pay attention to the initial conditions when solving for constants during integration in physics problems. They are crucial for determining the specific solution to the equations.