Let's solve the problem step by step.

Given:

1. Mass of the body ($m$) = $8 \, \text{kg}$
2. Acceleration as a function of time:
   $a(t) = 2t - 6 \, \text{m/s}^2$
3. Time interval: $[3, 5]$
4. We are asked to find the impulse of the force on the body over this interval.

Key Formula:

The impulse of a force is given by: $J = \Delta p = m \cdot \Delta v$ where $J$ is the impulse, $p$ is the momentum, $m$ is the mass, and $\Delta v$ is the change in velocity.

The change in velocity $\Delta v$ can be calculated by integrating the acceleration function: $v(t) = \int a(t) \, dt$

Step 1: Integrate $a(t)$ to find $v(t)$:

$a(t) = 2t - 6$ $v(t) = \int (2t - 6) \, dt = t^2 - 6t + C$ Here, $C$ is the constant of integration.

Step 2: Find $\Delta v$ over the interval $[3, 5]$:

The change in velocity is: $\Delta v = v(5) - v(3)$

Since the constant $C$ cancels out when calculating $\Delta v$, we only evaluate the definite integral of $a(t)$ over $[3, 5]$: $\Delta v = \int_{3}^{5} (2t - 6) \, dt$

Step 3: Compute the definite integral:

$\int (2t - 6) \, dt = \left[t^2 - 6t\right]_3^5$ Evaluate at the bounds: $v(5) = (5)^2 - 6(5) = 25 - 30 = -5$ $v(3) = (3)^2 - 6(3) = 9 - 18 = -9$ $\Delta v = v(5) - v(3) = (-5) - (-9) = 4 \, \text{m/s}$

Step 4: Calculate the impulse:

Using: $J = m \cdot \Delta v$ $J = 8 \cdot 4 = 32 \, \text{kg·m/s}$

Final Answer:

The impulse of the force on the body is: $\boxed{32 \, \text{kg·m/s}}$

Let me know if you'd like more details or have further questions!

Related Questions:

1. What is the physical meaning of impulse in terms of force and time?
2. How do you calculate acceleration if the velocity function is given instead?
3. How does changing the mass of the object affect the impulse in this scenario?
4. What would happen if the acceleration was a constant instead of time-dependent?
5. How can this problem be extended to include friction or air resistance?

Tip:

When solving impulse problems, always check if acceleration depends on time, as this often requires integration to find velocity or change in velocity.