The displacement of a particle starting from rest (at `t = 0)` is given by `s = 6t^(2) - t^(3)`. The time in seconds at which the particle will attain zero velocity again, is

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the problem, we need to find the time at which the particle attains zero velocity again after starting from rest. The displacement of the particle is given by the equation:

\[ s = 6t^2 - t^3 \]

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the time when the particle's velocity becomes zero again after starting from rest at \( t = 0 \).

2. **Find the Velocity**: The velocity \( v \) of the particle is the derivative of the displacement \( s \) with respect to time \( t \). We differentiate the given displacement function:

\[ v = \frac{ds}{dt} = \frac{d}{dt}(6t^2 - t^3) \]

3. **Differentiate the Displacement**:
- The derivative of \( 6t^2 \) is \( 12t \).
- The derivative of \( -t^3 \) is \( -3t^2 \).

So, the velocity function is:

\[ v = 12t - 3t^2 \]

4. **Set the Velocity to Zero**: To find when the particle attains zero velocity again, we set the velocity equation to zero:

\[ 12t - 3t^2 = 0 \]

5. **Factor the Equation**: We can factor out \( 3t \):

\[ 3t(4 - t) = 0 \]

6. **Solve for \( t \)**: Setting each factor to zero gives us:

- \( 3t = 0 \) which implies \( t = 0 \)
- \( 4 - t = 0 \) which implies \( t = 4 \)

Thus, the times at which the velocity is zero are \( t = 0 \) seconds and \( t = 4 \) seconds.

7. **Conclusion**: Since the question asks for the time at which the particle will attain zero velocity again, the answer is:

\[ t = 4 \text{ seconds} \]

### Final Answer:
The time in seconds at which the particle will attain zero velocity again is \( 4 \) seconds.

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