For motion of an object along x-axis, the velocity V depends on the displacement x as `V = 3x^(2) -2x`. Accelertion at `x= 2m` is 10n where n is an integer. The value of n is `"____________"`.

Explanation

To solve the problem, we need to find the acceleration of the object when the displacement \( x = 2 \, \text{m} \). The velocity \( V \) is given as a function of displacement \( x \):

\[ V = 3x^2 - 2x \]

### Step 1: Find the expression for acceleration
Acceleration \( a \) can be expressed in terms of velocity \( V \) and displacement \( x \) using the chain rule:

\[ a = V \frac{dv}{dx} \]

### Step 2: Differentiate the velocity function
We first need to differentiate the velocity function \( V \) with respect to \( x \):

\[ \frac{dV}{dx} = \frac{d}{dx}(3x^2 - 2x) \]

Using the power rule for differentiation:

\[ \frac{dV}{dx} = 6x - 2 \]

### Step 3: Substitute \( V \) and \( \frac{dV}{dx} \) into the acceleration formula
Now we can substitute \( V \) and \( \frac{dV}{dx} \) into the acceleration formula:

\[ a = (3x^2 - 2x)(6x - 2) \]

### Step 4: Substitute \( x = 2 \, \text{m} \) into the acceleration formula
Now, we will substitute \( x = 2 \) into the equation:

1. Calculate \( V \) at \( x = 2 \):

\[ V = 3(2^2) - 2(2) = 3(4) - 4 = 12 - 4 = 8 \]

2. Calculate \( \frac{dV}{dx} \) at \( x = 2 \):

\[ \frac{dV}{dx} = 6(2) - 2 = 12 - 2 = 10 \]

3. Now substitute these values into the acceleration equation:

\[ a = V \cdot \frac{dV}{dx} = 8 \cdot 10 = 80 \, \text{m/s}^2 \]

### Step 5: Relate acceleration to \( 10n \)
According to the problem, the acceleration is given as \( 10n \):

\[ 80 = 10n \]

### Step 6: Solve for \( n \)
Now, we can solve for \( n \):

\[ n = \frac{80}{10} = 8 \]

Thus, the value of \( n \) is:

\[ \boxed{8} \]