For motion of an object along the x-axis the velocity `v` dipends on the displacement `x` an `v=3x^(2)`, then what is the acceleration at `x=2 m`.

Options

A. `48 ms^(-2)`
B. `80 ms^(-2) m`.
C. `18 ms^(-2)`
D. `144 ms^(-2)`

Explanation

To find the acceleration of an object whose velocity depends on its displacement, we start with the given relationship between velocity \( v \) and displacement \( x \):

1. **Given Equation**:
\[ v = 3x^2 \]

2. **Acceleration Formula**:
The acceleration \( a \) can be expressed in terms of velocity and displacement using the chain rule:
\[ a = v \frac{dv}{dx} \]

3. **Finding \( \frac{dv}{dx} \)**:
To find \( \frac{dv}{dx} \), we differentiate \( v \) with respect to \( x \):
\[ \frac{dv}{dx} = \frac{d}{dx}(3x^2) = 6x \]

4. **Substituting \( v \) and \( \frac{dv}{dx} \) into the acceleration formula**:
Now we substitute \( v \) and \( \frac{dv}{dx} \) into the acceleration formula:
\[ a = v \cdot \frac{dv}{dx} = (3x^2)(6x) = 18x^3 \]

5. **Calculating acceleration at \( x = 2 \, \text{m} \)**:
Now we substitute \( x = 2 \, \text{m} \) into the acceleration equation:
\[ a = 18(2^3) = 18 \cdot 8 = 144 \, \text{m/s}^2 \]

Thus, the acceleration at \( x = 2 \, \text{m} \) is \( 144 \, \text{m/s}^2 \).