https://file.aiquickdraw.com/mathbot/file/17267918239613z8ecdq4.jpg

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.

Find the velocity and position vectors of a particle with a(t) = 8i + 3j, v(0) = k, and r(0) = i.

Learn how to find the velocity and position vectors of a particle with acceleration a(t) = 8i + 3j, initial velocity v(0) = k, and initial position r(0) = i. This solution involves vector calculus and integration, applying the fundamental theorem of calculus.

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