A body starts from the origin and moves along the X-axis such that the velocity at any instant is given by `(4t^3-2t)`, where t is in sec and velocity in m/s. what is the acceleration of the particle when it is 2 m from the origin?

Options

A. `28m//s^(2)`
B. `22 m//s^(2)`
C. `12 m//s^(2)`
D. `10 m//s^(2)`

Answer

To solve the problem step by step, we need to find the acceleration of a particle when it is 2 meters from the origin, given its velocity function.

### Step 1: Write down the velocity function
The velocity \( v \) of the particle is given by:
\[ v = 4t^3 - 2t \]

### Step 2: Find the acceleration function
Acceleration \( a \) is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Differentiating the velocity function:
\[ a = \frac{d}{dt}(4t^3 - 2t) = 12t^2 - 2 \]

### Step 3: Find the position function
To find the time when the particle is 2 meters from the origin, we need to integrate the velocity function. We know:
\[ v = \frac{dx}{dt} = 4t^3 - 2t \]
Integrating both sides with respect to \( t \):
\[ dx = (4t^3 - 2t) dt \]
Integrating:
\[ x = \int (4t^3 - 2t) dt = \left( t^4 - t^2 \right) + C \]
Since the particle starts from the origin, when \( t = 0 \), \( x = 0 \), we have \( C = 0 \). Thus:
\[ x = t^4 - t^2 \]

### Step 4: Set the position equal to 2 meters
We need to find \( t \) when \( x = 2 \):
\[ t^4 - t^2 = 2 \]
Rearranging gives:
\[ t^4 - t^2 - 2 = 0 \]

### Step 5: Substitute \( y = t^2 \)
Let \( y = t^2 \). Then the equation becomes:
\[ y^2 - y - 2 = 0 \]
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ y = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2} \]
This gives:
\[ y = 2 \quad \text{or} \quad y = -1 \]
Since \( y = t^2 \) cannot be negative, we take \( y = 2 \):
\[ t^2 = 2 \implies t = \sqrt{2} \text{ seconds} \]

### Step 6: Find the acceleration at \( t = \sqrt{2} \)
Now substituting \( t = \sqrt{2} \) into the acceleration function:
\[ a = 12t^2 - 2 = 12(2) - 2 = 24 - 2 = 22 \text{ m/s}^2 \]

### Final Answer
The acceleration of the particle when it is 2 meters from the origin is:
\[ \boxed{22 \text{ m/s}^2} \]