For a particle moving in a straight line, the displacement of the particle at time `t` is given by
`S=t^(3)-6t^(2) +3t+7`
What is the velocity of the particle when its acceleration is zero?

Options

A. `- 9 ms^(-1)`
B. `-12 ms^(-1)`
C. `3 ms^(-1)`
D. `42 ms^(-1)`

Explanation

To find the velocity of the particle when its acceleration is zero, we can follow these steps:

### Step 1: Write down the displacement equation
The displacement \( S \) of the particle at time \( t \) is given by:
\[ S = t^3 - 6t^2 + 3t + 7 \]

### Step 2: Find the velocity equation
The velocity \( V \) is the first derivative of displacement with respect to time:
\[ V = \frac{dS}{dt} = \frac{d}{dt}(t^3 - 6t^2 + 3t + 7) \]
Calculating the derivative:
\[ V = 3t^2 - 12t + 3 \]

### Step 3: Find the acceleration equation
The acceleration \( A \) is the derivative of velocity with respect to time:
\[ A = \frac{dV}{dt} = \frac{d}{dt}(3t^2 - 12t + 3) \]
Calculating the derivative:
\[ A = 6t - 12 \]

### Step 4: Set the acceleration to zero and solve for \( t \)
To find when the acceleration is zero, set the acceleration equation to zero:
\[ 6t - 12 = 0 \]
Solving for \( t \):
\[ 6t = 12 \\ t = 2 \text{ seconds} \]

### Step 5: Substitute \( t \) back into the velocity equation
Now that we have \( t = 2 \) seconds, we can find the velocity at this time:
\[ V = 3(2^2) - 12(2) + 3 \]
Calculating this:
\[ V = 3(4) - 24 + 3 \\ V = 12 - 24 + 3 \\ V = -9 \text{ m/s} \]

### Final Answer
The velocity of the particle when its acceleration is zero is:
\[ \text{Velocity} = -9 \text{ m/s} \]

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