To solve the problem step by step, we will use the work-energy principle and the relationship between force, mass, and acceleration.
### Step 1: Identify the given values
- Mass of the block, \( m = 2 \, \text{kg} \)
- Initial velocity, \( v_0 = 4\sqrt{5} \, \text{m/s} \)
- Force acting on the block, \( F = 3x^2 - 12x \, \text{N} \)
- Position where we need to find the velocity, \( x = 2 \, \text{m} \)
### Step 2: Write down the force equation
From Newton's second law, we know:
\[
F = ma
\]
where \( a \) is the acceleration. We can express acceleration in terms of velocity \( v \) and position \( x \):
\[
a = \frac{dv}{dt} = \frac{dv}{dx} \cdot v
\]
Thus, we can rewrite the force equation as:
\[
F = m \cdot v \cdot \frac{dv}{dx}
\]
### Step 3: Substitute the force into the equation
Substituting the expression for force into the equation gives:
\[
3x^2 - 12x = 2v \frac{dv}{dx}
\]
### Step 4: Rearranging and integrating
Rearranging the equation, we have:
\[
\frac{3x^2 - 12x}{2} = v \frac{dv}{dx}
\]
Now, we can separate variables:
\[
\int v \, dv = \int \frac{3x^2 - 12x}{2} \, dx
\]
### Step 5: Integrate both sides
Integrating the left side:
\[
\int v \, dv = \frac{v^2}{2} + C
\]
Integrating the right side:
\[
\int \frac{3x^2 - 12x}{2} \, dx = \frac{3}{2} \cdot \frac{x^3}{3} - \frac{12}{2} \cdot \frac{x^2}{2} = \frac{x^3}{2} - 3x^2
\]
### Step 6: Set up the limits for integration
We will evaluate from \( x = 0 \) to \( x = 2 \):
\[
\left[ \frac{v^2}{2} \right]_{v_0}^{v} = \left[ \frac{x^3}{2} - 3x^2 \right]_{0}^{2}
\]
### Step 7: Calculate the right side
Calculating the right side:
- At \( x = 2 \):
\[
\frac{2^3}{2} - 3(2^2) = \frac{8}{2} - 3(4) = 4 - 12 = -8
\]
- At \( x = 0 \):
\[
\frac{0^3}{2} - 3(0^2) = 0
\]
Thus, the right side evaluates to:
\[
-8 - 0 = -8
\]
### Step 8: Set up the equation
Now we have:
\[
\frac{v^2}{2} - \frac{(4\sqrt{5})^2}{2} = -8
\]
Calculating \( (4\sqrt{5})^2 = 80 \):
\[
\frac{v^2}{2} - \frac{80}{2} = -8
\]
This simplifies to:
\[
\frac{v^2}{2} - 40 = -8
\]
Thus:
\[
\frac{v^2}{2} = 32
\]
Multiplying both sides by 2 gives:
\[
v^2 = 64
\]
### Step 9: Solve for velocity
Taking the square root:
\[
v = \sqrt{64} = 8 \, \text{m/s}
\]
### Final Answer
The velocity of the block when it is at \( x = 2 \, \text{m} \) is \( 8 \, \text{m/s} \).
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