The displacement of a particle along an axis is described by equation `x = t^(3) - 6t^(2) + 12t + 1`.
The velocity of particle when acceleration is zero, is

Options

A. zero
B. -6
C. 12
D. 1

Explanation

To solve the problem, we need to find the velocity of the particle when its acceleration is zero. We will follow these steps:

### Step 1: Write down the displacement equation
The displacement of the particle is given by:
\[ x(t) = t^3 - 6t^2 + 12t + 1 \]

### Step 2: Find the velocity
The velocity \( v(t) \) is the first derivative of the displacement with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(t^3 - 6t^2 + 12t + 1) \]

Calculating the derivative:
\[ v(t) = 3t^2 - 12t + 12 \]

### Step 3: Find the acceleration
The acceleration \( a(t) \) is the derivative of the velocity:
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(3t^2 - 12t + 12) \]

Calculating the derivative:
\[ a(t) = 6t - 12 \]

### Step 4: Set the acceleration to zero
To find the time when the acceleration is zero, we set the acceleration equation to zero:
\[ 6t - 12 = 0 \]

Solving for \( t \):
\[ 6t = 12 \]
\[ t = 2 \, \text{seconds} \]

### Step 5: Find the velocity at \( t = 2 \)
Now we will substitute \( t = 2 \) into the velocity equation to find the velocity at that time:
\[ v(2) = 3(2^2) - 12(2) + 12 \]

Calculating:
\[ v(2) = 3(4) - 24 + 12 \]
\[ v(2) = 12 - 24 + 12 \]
\[ v(2) = 0 \]

### Final Answer
The velocity of the particle when the acceleration is zero is:
\[ \boxed{0} \]

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