To find the particle's velocity and position after 6.5 milliseconds, we need to analyze the given initial conditions and the acceleration function. The initial velocity is provided as \( v = (750 \, \text{cm/s}) \, \mathbf{j} \), which we can convert to meters per second for consistency: \( v = 7.5 \, \text{m/s} \, \mathbf{j} \). The acceleration is a function of time, given by \( \mathbf{a} = (15 \, \text{m/s}^6 \, t^4) \, \mathbf{i} - (10 \, \text{m/s}^2 + (8 \, \text{m/s}^5) \, t^3) \, \mathbf{j} + (35 \, \text{m/s}^4 \, t^2) \, \mathbf{k} \).
Step 1: Calculate Velocity
To find the velocity at a specific time, we integrate the acceleration function with respect to time. The velocity \( \mathbf{v}(t) \) can be expressed as:
\[
\mathbf{v}(t) = \mathbf{v}_0 + \int \mathbf{a}(t) \, dt
\]
Here, \( \mathbf{v}_0 = 0 \, \mathbf{i} + 7.5 \, \mathbf{j} + 0 \, \mathbf{k} \) is the initial velocity. We will integrate each component of the acceleration separately.
Integrating the Acceleration
- **i-component:**
Integrating \( 15 \, t^4 \) gives:
\[
\int 15 \, t^4 \, dt = 3 \, t^5 + C_1
\]
- **j-component:**
Integrating \( -10 - 8t^3 \) gives:
\[
\int (-10 - 8t^3) \, dt = -10t - 2t^4 + C_2
\]
- **k-component:**
Integrating \( 35t^2 \) gives:
\[
\int 35t^2 \, dt = \frac{35}{3}t^3 + C_3
\]
Now, substituting the constants of integration \( C_1, C_2, C_3 \) as zero (since the initial velocity at \( t = 0 \) is known), we have:
\[
\mathbf{v}(t) = \left(3t^5\right) \mathbf{i} + \left(7.5 - 10t - 2t^4\right) \mathbf{j} + \left(\frac{35}{3}t^3\right) \mathbf{k}
\]
Evaluating Velocity at \( t = 6.5 \, \text{ms} \)
Convert 6.5 milliseconds to seconds: \( t = 0.0065 \, \text{s} \). Now substitute \( t \) into the velocity equation:
- **i-component:**
\[
v_x(0.0065) = 3(0.0065)^5 \approx 0.0000000003 \, \text{m/s}
\]
- **j-component:**
\[
v_y(0.0065) = 7.5 - 10(0.0065) - 2(0.0065)^4 \approx 7.5 - 0.065 \approx 7.435 \, \text{m/s}
\]
- **k-component:**
\[
v_z(0.0065) = \frac{35}{3}(0.0065)^3 \approx 0.0000005 \, \text{m/s}
\]
Thus, the velocity at \( t = 6.5 \, \text{ms} \) is approximately:
\[
\mathbf{v}(0.0065) \approx (0.0000000003) \, \mathbf{i} + (7.435) \, \mathbf{j} + (0.0000005) \, \mathbf{k}
\]
Step 2: Calculate Position
Next, we find the position \( \mathbf{r}(t) \) by integrating the velocity function:
\[
\mathbf{r}(t) = \mathbf{r}_0 + \int \mathbf{v}(t) \, dt
\]
Assuming the particle starts at the origin, \( \mathbf{r}_0 = 0 \). We integrate each component of the velocity:
- **i-component:**
\[
\int 3t^5 \, dt = \frac{3}{6}t^6 = 0.5t^6
\]
- **j-component:**
\[
\int (7.5 - 10t - 2t^4) \, dt = 7.5t - 5t^2 - \frac{2}{5}t^5
\]
- **k-component:**
\[
\int \frac{35}{3}t^3 \, dt = \frac{35}{12}t^4
\]
Thus, the position function becomes:
\[
\mathbf{r}(t) = \left(0.5t^6\right) \mathbf{i} + \left(7.5t - 5t^2 - \frac{2}{5}t^5\right) \mathbf{j} + \left(\frac{35}{12}t^4\right) \mathbf{k}
\]
